The first indications of a non-trivial proton internal
structure were obtained in the the pioneering experiments of Hofstadter {\it et al.}
on elastic electron--nucleon scattering~\cite{Hofstadter:1955ae,Mcallister:1956ng}.
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In these experiments,
by examining the deviations from the simple Mott
scattering formulae for point--like particles, the finite extension
of the proton could be resolved.
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There, the charge radius of the proton was determined to be $\simeq 0.7$ fm to within a few percent precision.

 Although this result already hinted at an underlying substructure, the serious possibility that the proton is composite originated only later, with the ideas put forward independently by Zweig~\cite{Zweig:1964jf} and Gell--Mann~\cite{GellMann:1964nj} in 1964.
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By postulating the existence of three `aces' (Zweig's term) or `quarks' (Gell--Mann's) with fractional electric charge and baryon number, and spin--$1/2$, the complex structure of the hadrons and meson multiplets could be explained in a rather compact way.
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However, Zweig and Gell--Mann were understandably cautious about interpreting these objects as physical particles of finite mass, rather than simply convenient mathematical structures,
specially given that the mechanism for binding such quarks together was not understood,
and stable quarks had never been seen experimentally.

This situation changed dramatically in 1967 with the new experimental data on deep inelastic scattering (DIS)  provided by the SLAC 20 GeV linear accelerator.
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The physicists of the
SLAC--MIT collaboration were surprised to find that, in contrast to the case of elastic lepton--proton scattering, the two form factors associated with the DIS cross section, the so--called structure functions, were  roughly independent of $Q^2$~\cite{Bloom:1969kc,Breidenbach:1969kd}. Moreover, these appeared to exhibit the scaling behaviour predicted by Bjorken in 1969~\cite{Bjorken:1968dy}, namely that the structure functions should depend only on the ratio of $Q^2$ to the lepton energy loss $\nu$ in the proton rest frame\footnote{Indications of this scaling were also observed at the DESY electron synchrotron in the same year~\cite{Albrecht:1969zy}.}. 
 
 These observations led Feynman to introduce the parton model~\cite{Feynman:1969ej}, in which the incident lepton scatters incoherently and instantaneously from the point--like `partonic' constituents of the proton.
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This concept, developed further in~\cite{Bjorken:1969ja}, naturally explains the observed Bjorken scaling behaviour, with the point--like partons in this simple picture providing no additional scale through which Bjorken scaling could be broken. At the same time Callan and Gross~\cite{Callan:1969uq} showed that the DIS structure functions obey a simple relation for the case of spin--$\frac{1}{2}$ quark constituents, a finding that was also supported by the data~\cite{Miller:1971qb}.
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These partons were therefore naturally associated with the ``constituent''
quarks of Gell--Mann and Zweig.
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The subsequent demonstration of asymptotic freedom in 1973 in strongly interacting non--abelian gauge theories~\cite{Gross:1973id,Politzer:1973fx}  provided a simple explanation for the observed absence of free quarks, through the process of confinement, and the QCD parton model became the established approach to describe scattering processes in the strong interactions.
 
A central ingredient of this QCD parton model are the probability distributions of the momentum
carried by these partons, known as the
parton distribution functions, or
PDFs.
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The first studies concentrated on developing simple models for these objects based on the limited experimental input available, for example: in~\cite{Bjorken:1969ja} phase space considerations were used to conclude that the PDFs must also include a contribution from the now well-known sea of quark--antiquark pairs in addition to the valence quarks; in~\cite{Kuti:1971ph} a gluon PDF was introduced to account for the observed quark momentum fractions in a physically reasonable way consistent with energy conservation, and simple $x$ dependencies of the PDFs were predicted according to general Regge theory and phase space expectations. 

The idea of fitting a freely parameterised set of PDF followed soon after these initial studies.
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In~\cite{McElhaney:1973nj} the approach of~\cite{Kuti:1971ph} was extended to a more general phenomenological form, and a 4--parameter fit to the quark PDFs was performed to the available data on proton and neutron structure functions.
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As the amount and type of data increased, the shape of the PDFs became increasingly general, see for example~\cite{Barger:1973bn,Hinchliffe:1977jy}. Although the momentum fraction carried by the gluon could be determined by the missing contribution to DIS appearing in the momentum sum rule, it was only possible to fit its shape following the observation of scaling violations in the structure functions, first observed at FNAL~\cite{Fox:1974ry} in 1974.
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Such $Q^2$ dependent deviations from simple Bjorken scaling occur due to higher--order QCD corrections to DIS and were directly connected through the DGLAP equation~\cite{Altarelli:1977zs,gl,dok,Lipatov:1974qm,dok} to the $Q^2$ evolution of the PDFs. This allowed the first determinations of the shape of the gluon PDF to be made in~\cite{Gluck:1980cp} (see also~\cite{Baulieu:1978ze}).

The subsequent LO fits of~\cite{Eichten:1984eu}  (based on~\cite{Abramowicz:1982re}) to fixed target structure function and neutrino DIS data, and~\cite{Duke:1983gd}, which also included $J/\psi$ meson and muon pair hadroproduction, were widely used for a range of phenomenological applications. By the late 1980s, PDF fits at NLO in the strong coupling were standard, with the earlier analyses of~\cite{Martin:1987vw,Diemoz:1987xu} fitting to fixed target DIS and the subsequent fits of~\cite{Aurenche:1988vi,Harriman:1990hi,Morfin:1990ck} including prompt photon and Drell--Yan hadroproduction.
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Around the same time, the `dynamical' PDF set of~\cite{Gluck:1989ze} was produced with the assumption that at low scale the quark sea vanished and the gluon becomes proportional to the valence quark distributions, themselves determined from DIS data.
 
 Up to the early 90s, all DIS measurements was made with fixed target experiments and hence limited to the higher $x\gtrsim 0.01$ region.
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This changed in 1992 when the HERA high-energy collider at DESY started taking data.
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HERA collided 920 GeV protons with $\approx 27.5$ GeV electrons for most of the run period, allowing the previously unexplored region down to $x\sim 10^{-4}$ region to be probed at high $Q^2$.
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By 1994, data from HERA were included for the first in the MRS(A)~\cite{Martin:1994kn} and CTEQ3~\cite{Lai:1994bb} global fits. These were also the first fits to include data from the Tevatron $p\overline{p}$ collider, with in particular the $W$ asymmetry data providing new information on the quark flavour decomposition.
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In the years that followed, further public releases within these approaches were produced, including in particular the increasingly precise HERA measurements, as well as Tevatron data on jet production, which placed new and important constraints on the poorly-constrained medium and large-$x$ gluon. 
 
These PDF sets corresponded to the best fit only, that is, no estimate of the uncertainty on the PDFs due to the errors on the data in the fit were included, beyond relatively simple studies where a range of fits under different input assumptions might be performed to give some estimate of the spread.
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This was an acceptable situation when the uncertainties on the hadron collider data were sufficiently large, however as the data precision increased, the lack of a reliable
estimate of PDF uncertainties rapidly became an issue. In the 1996 CDF measurement~\cite{Abe:1996wy} of inclusive jet production, for example, there was an apparent excess of events at high jet $E_\perp$ that was interpreted at the time as a possible sign for new physics. In the subsequent study of~\cite{Huston:1995tw} it was shown that the gluon PDF could be modified in a way that still fit all available data, including the CDF jets. Clearly, a robust evaluation of the PDF uncertainties was needed to avoid such a situation.

The first attempts to produce such uncertainties, based on linear propagation of the experimental systematic and statistical errors through to the PDFs, considered a restricted set of DIS data~\cite{Alekhin:1996za,Botje:1999dj,Barone:1999yv,Giele:2001mr}. The extension of these methods to the wider data set included in a global PDF fit was a complicated problem, both from a purely technical point of view, but also more conceptually. In particular, more conventional statistical approaches to evaluating the uncertainty on the fitted PDF parameters, such as a standard `$\Delta \chi^2 =1$' variation, are only appropriate when fitting perfectly consistent data sets with purely Gaussian errors against a well--defined theory.
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For PDF fits based on a wide range of experimental measurements, none of these criteria are fulfilled: individual data sets are often found to have a low associated likelihood, with a $\chi^2$ per degree of freedom well above one, the experimental systematic uncertainties will not generally be Gaussian in nature, and the fixed-order perturbative theory calculation will carry its own (usually omitted) uncertainties.
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These issues were addressed in the CTEQ~\cite{Pumplin:2001ct,Pumplin:2002vw} and MRST~\cite{Martin:2002aw} PDF releases in 2002, with the basic idea being to allow the $\chi^2$ to vary from the minimum by a larger degree, or `tolerance', to account for the departure from the textbook statistical situation. 

The calculation of the NNLO splitting functions in 2004~\cite{mvvns,Vogt:2004mw} provided the necessary tools to go to NNLO in PDF fits, and with the release of the MSTW08~\cite{Martin:2009iq} and CT10~\cite{Gao:2013xoa} sets (the successors to the MRST and CTEQ sets, respectively) NNLO became the standard for global PDFs. At the same time the ABKM09~\cite{Alekhin:2009ni} NNLO PDFs were released. These were based on the earlier studies of~\cite{Alekhin:1996za,Alekhin:2000ch,Alekhin:2005gq}, and fit to a reduced data set of DIS and fixed target Drell--Yan and dimuon production, with a classical `$\Delta\chi^2=1$' error treatment applied. A further set to consider a reduced data sample to appear at this time was the HERAPDF1.0~\cite{Aaron:2009aa} PDFs. These included only the combined H1 and ZEUS measurements from the HERA Run I phase, with the aim of determining the PDFs from a completely consistent DIS data sample. This allowed the PDF uncertainty to again be described without the introduction of a larger tolerance factor, while the uncertainties due to model assumptions and choice of parameterisation were included in addition. This NLO set was extended to NNLO in the HERAPDF2.0~\cite{Abramowicz:2015mha} PDFs, which used the final combined HERA I + II data sample. The NNLO JR09~\cite{JimenezDelgado:2008hf} set included a range of DIS and fixed target data, applying both a `standard' fitting approach and the `dynamical' approach of~\cite{Gluck:1989ze}. The subsequent JR14~\cite{Jimenez-Delgado:2014twa} set included a range of data updates, including jet production from the Tevatron.

The approaches described above differ greatly in many respects, both in the choice of input data sets, and the treatment of the corresponding theory predictions. However, while there are significant differences in the precise choice of parameterisation, in all cases these rely on parameterising the PDFs in terms of reasonably contained, $O(20-40)$, number of free variables. Moreover, while the precise prescription may vary, these are again all based on the `Hessian' linear error propagation procedure. A different approach, first discussed in~\cite{Forte:2002fg}, was taken by the NNPDF collaboration.
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There, the PDF functional forms are based on artificial neural networks, in particular
feed-forward multi-layer perceptrons, allowing many more, $O(200-300)$, free parameters. In addition, rather than constructing the PDF error from the $\chi^2$ variation about the best fit values, a `Monte Carlo' (MC) approach is taken, with a large enough sample of PDF `replica' sets each fit to randomly distributed pseudo--data generated according to the measured data values and their uncertainties. The first NNPDF1.0 fit was reported in~\cite{Ball:2010de}, at NLO and to a range of DIS and fixed target data. Subsequently, NNPDF2.1~\cite{Ball:2011mu,Ball:2011uy} provided the first NNLO PDF set within this approach, and included Tevatron data for the first time. 

More recently, data from the LHC has played an increasingly important role in PDF determination. The CT14~\cite{Dulat:2015mca}, MMHT14~\cite{Harland-Lang:2014zoa} and NNPDF2.3~\cite{Ball:2012cx} sets included LHC data on jets, $W$ and $Z$ boson production, and top pair production for the first time. In addition, ABM12~\cite{Alekhin:2013nda} was the first set from this group to include constraints from the
LHC, with data on $W$ and $Z$ boson production and top quark pair production.
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As we will discuss in some detail in this review, these
LHC measurements, which are being produced with increasingly high precision,
are now providing some of the most stringent constraints on the PDFs.

In parallel to these developments, there has been increasing focus on the use of PDFs as precise tools for LHC phenomenology, emphasising the need for clear benchmarking exercises between sets and PDF combinations, to provide an overall PDF uncertainty.
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The PDF4LHC Working Group, created in 2008, has played a significant role in this, with the benchmarking described in~\cite{Alekhin:2011sk} leading to first so--called PDF4LHC recommendation~\cite{Botje:2011sn} for the use of PDFs and their uncertainties at the LHC.
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This has subsequently been updated in~\cite{Butterworth:2015oua} (see~\cite{Accardi:2016ndt} for an alternative point of view).
 
