As discussed in Sect.~\ref{sec:fitmeth}, the NNPDF fitting methodology is based on  the
combination of three main components: i) the use of artificial neural networks
as universal unbiased interpolants, ii) the Monte Carlo method to estimate
and propagate PDF uncertainties, and iii) Genetic Algorithms (GA)
minimization for the training of
the neural networks combined with a look-back cross-validation stopping
to avoid over-fitting.
%
Here we review the historical development of the NNPDF family of PDF fits,
highlighting recent progress.

The NNPDF methodology was first presented in~\cite{Forte:2002fg}, where it was used
to produce a neural network based determination of the proton, deuteron
and non-singlet DIS structure functions from the fixed target data from NMC and BCDMS.
%
As a first phenomenological application, this determination was used
to extract the strong coupling constant $\alpha_S(m_Z)$ from scaling violations
of truncated moments of structure functions~\cite{Forte:2002us}.
%
This analysis was subsequently extended~\cite{DelDebbio:2004qj} to include as well
the $F_2^p$ measurements from the H1 and ZEUS experiments at the HERA collider.
%
Note that a determination of structure functions is purely data-driven, with no theoretical
input required at this point.

When moving from fitting structure functions to PDFs, there are a number of simplifications,
for instance one needs to fit only a 1D function $q_i(x,Q_0)$ as opposed to a 2D function $F_2^p(x,Q^2)$,
but also technical complications, the most important one being able to compute
DIS structure functions starting from the neural-network based parametrization of $q_i(x,Q_0)$.
%
First of all, the usual ANN training algorithm of back-propagation cannot be used in this
case, due to the convolution of the PDFs with the DGLAP evolution kernels and the
DIS coefficients functions.
%
To overcome this limitation, it was demonstrated how Genetic Algorithms can be efficiently
used for ANN training under a non-trivial mapping between the latter and the experimental data,
and used to extract the QCD vacuum condensates from hadronic tau decay data~\cite{tau}.
%
An efficient method to solve the DGLAP evolution equations in $N$-space was also developed,
called the {\tt Fast Kernel} method.
%
With these ingredients at hand, it became possible for the first time to apply the
NNPDF methodology to a determination of parton distributions, starting from
a fit of the non-singlet combination $q_{NS}(x,Q_0)$~\cite{DelDebbio:2007ee} and them moving to a first
full-fledged NLO PDF fit based on neutral-current DIS structure function data~\cite{Ball:2008by} in 
the NNPDF1.0 analysis.

Subsequently, the global NNPDF fits were improved both by adding new experimental data,
updating the theoretical calculations and/or refining the fitting methodology.
%
The NNPDF1.2 analysis~\cite{Ball:2009mk}
relaxed the previous assumption that
the strange sea was proportional to the light quark sea, $s=\bar{s}=\kappa \lp \bar{u}
+\bar{d}\rp$, and parametrized both $s^+$ and $s^-$ using neural networks, exploiting
the constraints from the NuTeV dimuon charged-current neutrino scattering data.
%
Two important phenomenological consequences of this analysis were, first of all, the demonstration
that the PDF uncertainties associated with $s^-$ were enough to remove the
NuTeV anomaly~\cite{Davidson:2001ji}
in the determination of the weak mixing angle $\sin^2\theta_W$; and second,
a direct extraction of the CKM matrix element $V_{cs}$ with a precision compatible
with that of the PDG average.

In 2010, the NNPDF2.0 set was released~\cite{Ball:2010de},
which constituted the first truly global PDF fit from the NNPDF collaboration.
%
In addition to the NC and CC DIS structure function data included in previous releases, NNPDF2.0
included in addition fixed-target Drell-Yan cross-sections from the Fermilab E605
and E866 experiments, inclusive jet production measurements from CDF and D0 and the Tevatron
as well as the differential rapidity distributions of the $Z$ boson also from
the Tevatron.
%
From the theoretical point of view, NNPDF2.0 was still based on the zero-mass VFN
scheme, and thus charm and bottom structure function data from HERA were not included.
%
A good overall description of all experiments in the global fit was found.
%
NNPDF2.0 was also the first global PDF set to include the recently release HERA combination
of H1 and ZEUS structure function data for the Run I data period~\cite{Aaron:2009aa}.

While NNPDF2.0 demonstrated that the NNPDF methodology could be successfully applied
to a global determination of parton distributions, there were still a number of important
limitations from the theoretical point of view.
%
First, the use of a ZM-VFN scheme neglected heavy quark mass effects in the DIS
structure functions, which were known to be important for the
description of the low-$x$, low-$Q^2$ HERA data.
%
Second, all NNPDF fits so far were based on NLO theory, and NNLO accuracy was essential
to match the corresponding precision of important partonic hard-scattering cross-sections
such as Higgs production in gluon fusion.
%
The first of these theory limitations was removed with the release of NNPDF2.1~\cite{Ball:2011mu},
which was based on the FONLL general-mass VFN for the calculation of DIS structure functions, which
allowed the HERA charm and bottom structure functions data to be fit.
%
This analysis also showed that the impact of heavy quark mass effects was less drastic
then previously reported, with the cross-section predictions between NNPDF2.0 and 2.1 typically
agreeing at the one-sigma level.
%
The NNPDF2.1 fit was also used to produce a determination of the strong coupling $\alpha_s(m_Z)$
from the global dataset~\cite{Lionetti:2011pw}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.45]{5-pdfgroups/plots/20momsumrule.pdf}
   \caption{\small 
   Left plot: comparison between the NNPDF2.0 and 2.1 predictions
   for the inclusive $W^+$ production cross-section at the LHC 7 TeV,
   which illustrates the phenomenological impact of heavy quark mass effects.
   %
   Right plot: the distribution of the momentum integral Eq.~(\ref{eq:momentumintegral})
   among the MC replicas for the variants of the NNPDF2.1 LO, NLO, and NNLO fits that
   do not impose the momentum sum rule.
    \label{fig:20momsumrule}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The second of these theoretical limitations was removed shortly after, with the
release of a NNLO version of NNPDF2.1~\cite{Ball:2011uy}.
%
This PDF set was based on the same dataset as its NLO counterpart, but with the DIS and hadronic cross-sections computed at NNLO, in the former
case using the FONLL-C GM-VFN scheme.
%
In the same publication, the first NNPDF LO sets were also presented.
%
The availability of NNPDF2.1 fits at LO, NLO and NNLO allowed a systematic
study of the perturbative convergence of the global fit, finding in particular reasonable
agreement at the one-sigma level between the NLO and NNLO versions.
%
The  consistency of the global QCD analysis framework was also tested by performing
fits without imposing the momentum sum rule and then verifying a posteriori that
the global fit result was consistent with the QCD expectation, finding indeed that
at NNLO
\be
\label{eq:momentumintegral}
\lc M \rc \equiv \int_0^1 dx\,\lp g(x,Q^2)+\Sigma(x,Q^2)\rp=1.002\pm 0.014 \, .
\ee
%
The NNPDF2.1 NNLO analysis was also used to perform a determination of
the strong coupling constant~\cite{Ball:2011us}, finding a value
$\alpha_s(m_Z)=0.1173\pm 0.0007^{\rm stat}\pm 0.0009^{\rm pert}$, a result which is 
included in the current PDG global average of $\alpha_s$~\cite{Olive:2016xmw}.

In the same way than in the early 90s the availability of the HERA structure function measurements
became a game--changer for global fits, from 2010 the LHC experiments started
producing a wealth of PDF-sensitive information for global fits.
%
With this motivation, in 2012 the NNPDF2.3 set was released~\cite{Ball:2012cx},
and was the first
PDF fit to include LHC data, in particular electroweak gauge boson production
from ATLAS, CMS and LHCb as well as jet production from ATLAS.
%
As with all subsequent releases, NNPDF2.3 was available at LO, NLO and NNLO
and was based on the FONLL general mass scheme.
%
The NNPDF2.3 set became the baseline PDF set in several popular Monte Carlo
event generators, such as {\tt Pythia8}~\cite{Sjostrand:2014zea}
and {\tt aMC@NLO}~\cite{Alwall:2014hca}.
%
Based on NNPDF2.3, an study of theoretical uncertainties affecting the
PDF from sources such as deuteron corrections and higher twists was
presented in~\cite{Ball:2013gsa}. 

Following the release of NNPDF2.3, it was realized that the increase in complexity required
to include the many new experiments that were either available or about to be released
could not be satisfactory tackled with the current, {\tt FORTRAN77}-based code.
%
With this motivation, a complete rewriting of the NNPDF global analysis framework
into {\tt C++} and {\tt Python} was undertaken, a two-year long effort that
culminated with the release of the NNPDF3.0 set~\cite{Ball:2014uwa}.
%
In addition to including many new LHC experiments on jets, vector boson production,
$W$+charm and top production, the main result of NNPDF3.0 was the systematic validation
of the complete fitting methodology based on statistically robust closure tests.

In these closure tests, pseudo-data was generated based on some ``truth'' PDFs, and then
a PDF fit was performed: if the resulting PDF central values and uncertainties
were consistent with the (known) input PDFs, the the closure test can be considered successful.
%
In Fig.~\ref{fig:nnpdf30closuretests} we show 
representative results of the closure tests presented
   in the NNPDF3.0 analysis.
   %
   In the left plot we show the
   results of a level 0 closure test, where the pseudo-data is generated
   without any statistical fluctuations, the $\chi^2$ should decrease monotonically
   as a function of the number of GA iterations, down to arbitrarily small values.
   %
   And in the right plot we show
   the distribution of the difference between theory and data,
   in units of the error of the latter among each of the Monte Carlo replicas.
   %
   This is consistent with the expectations from an underlying
   Gaussian distribution.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.47]{5-pdfgroups/plots/nnpdf30closuretests.pdf}
   \caption{\small Representative results of the closure tests presented
   in the NNPDF3.0 analysis.
   %
   Left plot: in a level 0 closure tests, where the pseudo-data is generated
   without any statistical fluctuations, the $\chi^2$ should decrease monotonically
   as a function of the number of GA iterations, down to arbitrarily small values.
   %
   Right plot: the distribution of the difference between theory and data
   in units of the error of the latter among each of the Monte Carlo replicas.
   %
   This distribution is consistent with the Gaussian expectations.
    \label{fig:nnpdf30closuretests}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A recent development in the NNPDF family of global analyses concerns the treatment of
the charm PDF.
%
In previous PDF sets, it was assumed that the charm
PDF $c(x,Q)$ was generated dynamically from the gluons and light quarks, as dictated by the DGLAP
evolution starting from the charm mass threshold $\mu_c\simeq m_c$.
%
However, a possible non-perturbative component of the charm PDF would invalidate this assumption,
a hypothesis which can ultimately be tested against experimental data.
%
In addition, treating the charm PDF on an equal footing with the gluon and light quark
PDFs offers other potential advantages, such as a reduced dependence on the value of
$m_c$ and an improved  agreement between data and theory from the more flexible input PDF
parametrization.

Therefore, a variant of the NNPDF3.0 fit with a fitted charm PDF was studied
in~\cite{Ball:2016neh}.
%
By parametrising the charm PDF with an artificial neural network with 37 free parameters, we found that
fitting charm leads to an improved $\chi^2$ for several experiments, stabilized the dependence
of the fit with respect to the value of $m_c$ and moreover allowing 
a satisfactory description, $\chi^2/N_{\rm dat}\simeq 1$, of the EMC charm structure function
data.
%
The resulting charm PDF can be compared with non-perturbative models~\cite{Brodsky:1980pb},
and some tantalizing evidence
for a large-$x$ non-perturbative charm component in the proton was found.
%
Predictions for a number of LHC process such as $Z+$charm and large-$p_T$ $D$ meson production
were also provided, showing the potential of Run II data to disentangle the charm
content of the proton.
%
See Sect.~\ref{sec:structure.charm} for more details about the charm
content of the proton.

The most recent incarnation of the NNPDF global analysis is the 
NNPDF3.1 set~\cite{Ball:2017nwa}.
%
The main motivation for this release was the availability of a large number of high-precision
collider measurements (and the corresponding
NNLO calculations) providing PDF-sensitive information, including some 
that for the first time could be used in a PDF fit, such as differential
distributions in top quark pair production and the $p_T$ of $Z$ bosons.
%
The second main motivation was to provide a state-of-the-art PDF set without assuming
that charm is generated perturbatively, that is, providing baseline global
PDF fits with fitted charm.
%
Some of the new experiments included in NNPDF3.1 were the $t\bar{t}$ distributions
from ATLAS and CMS, the legacy LHCb inclusive $W$ and $Z$ measurements from Run II,
the D0 $W$ asymmetries in the muon and electron channel, the $p_T$ of $Z$ bosons
from ATLAS and CMS at $\sqrt{s}=8$ TeV, as well as several other inclusive gauge boson
and jet production measurements from ATLAS and CMS.
