\label{sec:introduction}

The determination of the quark and gluon structure of the proton
is a central component
of the precision phenomenology program at the Large Hadron Collider (LHC).
%
This internal structure is quantified in the collinear QCD factorization framework by the
Parton Distribution Functions (PDFs), which encode the information related to
the momentum distribution of quarks and gluons within the proton.
%
Being driven by low-scale non-perturbative dynamics,
PDFs cannot be computed from first principles, at least
with current technology, and therefore they
need to be determined using experimental data from
a variety of hard-scattering cross-sections in
lepton-proton and proton-proton collisions.
%
This global QCD analysis program involves combining the most PDF-sensitive
data and the highest precision QCD and electroweak calculations available within a statistically
robust fitting methodology.
%
See Refs.~\cite{Rojo:2015acz,Butterworth:2015oua,Ball:2012wy,
Alekhin:2011sk,Forte:2013wc,Forte:2010dt,Perez:2012um,DeRoeck:2011na,
Accardi:2016ndt} for recent reviews on PDF determinations.

One of the main motivations to improve our understanding of the internal
structure of the proton is provided by the fact that
PDFs and their associated uncertainties play a decisive role in
several LHC applications.
%
To begin with, they represent one of the dominant theoretical uncertainties
for the determination of the Higgs boson couplings~\cite{deFlorian:2016spz}, where any deviation
from the tightly fixed SM predictions would indicate a smoking gun for new physics.
%
PDF uncertainties also affect the production of new high-mass resonances,
as predicted by many Beyond the Standard Model (BSM) scenarios~\cite{Beenakker:2015rna}, since
they probe PDFs at large values of the momentum fraction $x$, for which
current experimental constraints are scarce.
%
A third example is provided by the measurement of precision SM parameters at hadron
colliders, such as the $W$ mass~\cite{Bozzi:2015hha,Bozzi:2015zja,Bozzi:2011ww,Aaboud:2017svj}
or the strong coupling constant $\alpha_s(Q)$~\cite{,Khachatryan:2014waa,Aaboud:2017fml,
Chatrchyan:2013txa,Rojo:2014kta,Chatrchyan:2013haa}.
%
These can be sensitive to BSM effects (for instance via virtual effects of new
particles too heavy to be produced directly~\cite{Becciolini:2014lya,Dimopoulos:1981yj})
and in many cases PDF uncertainties
represent one of the limiting factors of the measurements.

Beyond the LHC, there are also several other instances where PDFs play an important
role, for instance in astroparticle physics, such as for the accurate
predictions for signal~\cite{CooperSarkar:2011pa}
and background~\cite{Gauld:2015kvh,Garzelli:2016xmx,Zenaiev:2015rfa,Gauld:2016kpd}
events at ultra-high energy neutrino telescopes.
%
And needless to say, parton distributions will keep playing an important role
for any future higher-energy collider involving hadrons in the
initial state~\cite{Mangano:2016jyj,AbelleiraFernandez:2012cc},
and therefore improving our understanding of PDFs also
strengthens the physics potential
of such future colliders.

A number of collaborations provide regular updates of their PDF sets, see~\cite{Ball:2014uwa,Dulat:2015mca,
  Harland-Lang:2014zoa,Alekhin:2017kpj,Abramowicz:2015mha,Jimenez-Delgado:2014twa,Accardi:2016qay}
  and references therein.
%
There are a range of differences between these analyses, arising for example at the level of the selection of the input fitted dataset,
the theoretical calculations of cross-sections, methodological choices
for the parametrization of PDFs, the estimate and propagation of PDF uncertainties, and the treatment
of external parameters.
%
For instance, while some PDF fits are based on a global dataset, including the widest possible
variety of experimental constraints, some others are based instead
on reduced datasets (for example, without
jet data) or even on a single dataset, as the HERAPDF2.0 set~\cite{Abramowicz:2015mha}
which is based entirely on the HERA inclusive
structure functions.


Despite these differences, it has been shown that, under some well-specified conditions,
PDF sets can be statistically combined among them into a unified set.
%
The most popular realization of this paradigm are the PDF4LHC15 sets~\cite{Butterworth:2015oua},
which combine the CT14, MMHT14, and NNPDF3.0 sets using the Monte Carlo (MC) method~\cite{Watt:2012tq},
and are subsequently reduced to small number of Hessian eigenvectors~\cite{Gao:2013bia,Carrazza:2015aoa} or MC
replicas~\cite{Carrazza:2015hva} to facilitate phenomenological applications.

This Report is motivated by the fact that the recent years have seen a number of important breakthroughs
in our understanding of the quark and gluon structure of the proton.
%
To begin with, the impressive recent progress in NNLO QCD calculations has now made it possible
to include essentially all relevant collider cross-sections consistently into
a NNLO global analysis, from top-quark differential distributions~\cite{Czakon:2016dgf} to inclusive
jets~\cite{Currie:2016bfm} and dijets~\cite{Currie:2017eqf},
isolated photons~\cite{Campbell:2016lzl}, and the $p_T$ distribution of
$Z$ bosons~\cite{Boughezal:2015dva,Ridder:2016nkl}, among others.
%
These theoretical developments
have been matched by the availability of high-precision measurements
from ATLAS, CMS, and LHCb at different center-of-mass energies
$\sqrt{s}$, in several cases with statistical uncertainties
at the per-mile level and systematic errors at the few-percent level.
%
The combination of these state-of-the art calculations and high-precision
data provides an unprecedented opportunity to constrain
the PDFs, but it also represents a challenge to verify if the global
QCD analysis framework can satisfactorily accommodate them.
%
Indeed, the validity of the QCD factorization theorems is being pushed to a level
never reached before.

In addition, there are a number of important topics that have attracted a lot
of attention recently.
%
One of these
is the role that QED and electroweak effects, and specifically
the photon PDF $\gamma(x,Q^2)$, play in global fits of parton distributions.
%
Recent progress has demonstrated that the photon PDF can be computed with
percent level accuracy~\cite{Manohar:2016nzj,Manohar:2017eqh},
improving on existing model~\cite{Martin:2004dh,Schmidt:2015zda,Harland-Lang:2016kog} and data-driven
determinations~\cite{Giuli:2017oii,Ball:2013hta,Bertone:2016ume}, with direct implications
for LHC cross-sections.
%
Another development is the treatment of the charm quark PDF on an equal
footing to the light quarks in global fits~\cite{Ball:2016neh}, which
stabilises the $m_c$ dependence, improves the agreement with high-precision data,
and allows a direct comparison with non-perturbative models of the charm
content of the proton~\cite{Brodsky:2015fna,Hou:2017khm}.
%
From the methodological point of view, there have been several improvements in the way that
PDFs are parametrized and the various associated sources of uncertainty are estimated among the
PDF fitting groups.
%
In addition, there has also been a recent explosion in the number of tools available for PDF studies,
from the open-source fitting framework {\tt xFitter}~\cite{Alekhin:2014irh}, to new
fast (N)NLO interfaces and public codes for the PDF
evolution~\cite{Bertone:2013vaa,Salam:2008qg,Botje:2010ay} and the
efficient calculation of hadronic
cross-sections~\cite{Carli:2010rw,Bertone:2016lga,amcfast,Wobisch:2011ij,Czakon:2017dip}.
%
It is therefore the goal of this Report to present a detailed overview of these
various recent developments, and how they have modified our present understanding
of the quark and gluon structure of the proton, with emphasis
on the resulting phenomenological applications.

This Report focuses only on one of the main aspects of the internal structure of
nucleons, namely collinear unpolarized PDFs, which are its
most relevant feature for the exploration of the high-energy frontier at the LHC.
%
There are however many other aspects of the inner life of
protons that would each deserve their own separate review, and that
due to space limitations cannot be covered here.
%
These include, among others,
the determination of
its spin structure by means of the polarized
PDFs~\cite{Nocera:2014gqa,deFlorian:2009vb}, the nuclear modifications
of the free-proton PDFs~\cite{Eskola:2016oht,Kusina:2016fxy}, relevant for the understanding of cold nuclear matter effects
at the  RHIC and LHC heavy-ion program, and 
the three-dimensional
imaging of nucleons in terms of transverse-momentum-dependent
PDFs (TMD-PDFs)~\cite{Angeles-Martinez:2015sea}.
%
We only note here that these fields do not exist in isolation, and progress in these topics
can often directly impact unpolarized PDF fits, and vice versa.

The structure of this Report is as follows.
%
First, in Sect.~\ref{sec:globalqcd}
we review the theoretical foundations
of the global PDF analysis framework, specifically
the QCD factorization theorems
of lepton-hadron and hadron-hadron collisions, as well
as the scale dependence of the PDFs as encoded in the DGLAP equations.
%
In Sect.~\ref{sec:datatheory}  we discuss the hard-scattering
experimental data, as well as the corresponding
state-of-the-art theoretical calculations, that are used
to constrain the PDFs in modern global analyses.
%
In Sect.~\ref{sec:fitmeth} we present
the methodological framework of PDF fits, including the various
approaches to parametrizing the PDFs and estimating and propagating
the uncertainties from theory and data to physical
cross-sections.
%
In Sect.~\ref{sec:pdfgroups} we summarize the main features
of the different PDF collaborations that provide regular updates
of their PDF fits, and then in Sect.~\ref{sec:structure} we compare
them, assessing their differences and similarities for different
aspects of the proton structure such as the gluon PDF, quark-flavour
separation, and the strange and charm content of the proton.

In Sect.~\ref{sec:QED} we then discuss the role
that QED and electroweak corrections play in modern PDF fits,
with emphasis on the photon content of the proton.
%
In Sect.~\ref{sec:LHCpheno} we highlight a number of
representative examples of the role of PDFs and their
uncertainties for the LHC precision physics program, in particular
the characterization of the Higgs sector, searches for massive New Physics resonances,
and the measurement of precision parameters such as $M_W$.
%
In the last part of this Report, Sect.~\ref{sec:PDFfuture},
we discuss some of the topics that are likely
to play an important role for the future of PDF determinations,
such as the quantification of theoretical
uncertainties, the interplay with lattice QCD calculations,
and the application of PDFs for future higher energy lepton-proton
and proton-proton colliders.
%
Finally, we conclude in Sect.~\ref{sec:conclusions}.
