\label{sec:conclusions}

The wealth of experimental data collected by the LHC experiments
so far represents only a small fraction of the complete
dataset that it will provide over the coming two decades of planned operations.
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This will allow the SM to be studied and stress-tested to an even greater, and completely unprecedented, level.
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Together with recent progress in theoretical calculations, we are therefore entering the era of
the LHC precision physics, which will aim to compare data and theory
at the percent level or even less.
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Given the null results of BSM searches so far, a systematic high-precision
analysis of the SM predictions and the LHC data might be one of most promising
approaches to look for BSM dynamics at the LHC, which may only be
manifest as subtle differences with the SM predictions.
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The detailed mapping of the quark and gluon structure of the
proton represents then a crucial component of this LHC precision physics
program.

In this Report we have presented an overview of the most important recent
developments in PDF determinations, with emphasis for their implications for
LHC phenomenology.
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After a succinct review of the theoretical foundations of the global QCD analysis framework,
we have reviewed recent progress both from the theoretical and the experimental
point of view for those hard-scattering cross-sections used in PDF fits; we have
compared the similarities and differences between the methodologies uses
for the various PDF fitting collaborations; and then presented the state-of-the-art
fits from each group and assessed what we can learn about the internal structure of the
nucleons from various points of view.
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We have then discussed the role of QED corrections in PDF fits, in particular concerning
photon-initiated processes, and presented some of the most representative
examples of the applications of PDFs for LHC phenomenology, from the measurement
of the Higgs boson couplings to the determination of the mass of the $W$ boson.

In the last section of these Report we have attempted to speculatively discuss some topics
that very likely will play a crucial role in the near-term future of PDF determinations.
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One of these topics is the role of theoretical uncertainties, in particular those arising from
missing higher-order terms in the perturbative expansion.
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Given the size of PDF uncertainties in the current generation of global analyses, it is likely that these theory
errors will increasingly become comparable (if not larger) than the nominal PDF uncertainties, and thus
finding a statistically sound method to account for these is of utmost importance.
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Another topic that might affect the PDF fitting paradigm is that of the interplay with lattice
QCD calculations, where recent progress both in computing Mellin moments of various flavor
combinations as well in direct $x$-space calculation of PDFs suggest that in the future the constraints
from non-perturbative
lattice calculations might be able to provide information for global PDF fits.
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Finally, we have summarized the importance of PDFs for future higher-energy colliders whose
physics case is being discussed at present, such as a Large Hadron electron Collider or a new
proton-proton collider with a center-of-mass energy of up to 100 TeV.

We hope that this Report has managed to convey to the reader that the topic of PDF determinations
is an interesting and lively one, with implications from the understanding of the non-perturbative
dynamics of the strong interactions to searches for new BSM physics and ultra-high energy astrophysics.
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In some respect, PDF fits represent a unique stress-test of the SM and of the collinear
QCD factorization framework.
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The latter appears to be in remarkably
good shape, given that we are now
able to simultaneously describe a few tens of individual experiments spanning a wide
kinematical domain, some of them with extremely
small uncertainties at the per-mile level.


PDF fits thrive at the cross-roads of advanced data analysis, state-of-the-art perturbative
calculations, and modern robust statistical methodology, and thus provide guidance for
other similar global analyses efforts, such as fits of the Wilson coefficients within
the SM Effective Field Theory framework.
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Moreover, progress in unpolarized PDF fits is also one of the main drivers of recent
improvements of other related aspects of the proton structure, from
polarized PDFs to nuclear PDFs, as well as to other aspects of non-perturbative QCD
such as light and heavy hadron fragmentation.

As we enter in the LHC precision era, ever-improving PDF determinations will keep providing a unique
contribution to
this exciting exploration of the high-energy frontier. 
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We fully expect that the requirements of
this LHC precision program will further drive improvements in global PDF determinations,
leading to an ever more detailed picture of the inner life of the proton.
