  Within the theory of Quantum Chromodynamics (QCD), the rich structure of
hadrons can be quantitatively characterized, among others, using a basis of
universal non-perturbative functions: parton distribution functions (PDFs),
generalized parton distributions (GPDs), transverse-momentum dependent
distributions (TMDs) and distribution amplitudes (DAs). For more than half a
century, there has been a joint experimental and theoretical effort to obtain
these partonic functions. However, the complexity of the strong interactions
has placed severe limitations, and first-principle results on the distributions
was extracted mostly from their moments computed in Lattice QCD. Recently,
breakthrough ideas changed the landscape and several approaches were proposed
to access the distributions themselves on the lattice.
  In this paper, we review in considerable detail approaches directly related
to partonic distributions. We highlight a recent idea proposed by X. Ji on
extracting quasi-distributions that spawned renewed interest in the whole field
and sparked the largest amount of numerical studies of Lattice QCD. We discuss
theoretical and practical developments, including challenges that had to be
overcome, with some yet to be handled. We also review numerical results,
including a discussion based on evolving understanding of the underlying
concepts and theoretical and practical progress. Particular attention is given
to important aspects that validated the quasi-distribution approach, such as
renormalization, matching to light-cone distributions and lattice techniques.
  In addition to a thorough discussion of quasi-distributions, we consider
other approaches: hadronic tensor, auxiliary quark methods, pseudo-PDFs, OPE
without OPE and good lattice cross sections. In closing, we provide prospects
of the field, with emphasis on the necessary conditions to obtain results with
controlled uncertainties.
