We review the basic theory of the parton pseudodistributions approach and its applications to lattice extractions of parton distribution functions. The crucial idea of the approach is the realization that the correlator $M(z,p)$ of the parton fields is a function ${cal M} ( u, -z^2)$ of Lorentz invariants $ u =-(zp)$, the Ioffe time, and the invariant interval $z^2$. This observation allows to extract the Ioffe-time distribution ${cal M} ( u, -z^2)$ from Euclidean separations $z$ accessible on the lattice. Another basic feature is the use of the ratio ${mathfrak M} ( u,-z^2) equiv {cal M} ( u, -z^2)/{cal M} (0, -z^2)$, that allows to eliminate artificial ultraviolet divergence generated by the gauge link for space-like intervals. The remaining $z^2$-dependence of the reduced Ioffe-time distribution ${mathfrak M} ( u,-z^2) $ corresponds to perturbative evolution, and can be converted into the scale-dependence of parton distributions $f(x,mu^2)$ using matching relations. The $ u$-dependence of ${mathfrak M} ( u,-z^2) $ governs the $x$-dependence of parton densities $f(x,mu^2)$. The perturbative evolution was successfully observed in exploratory quenched lattice calculation. The analysis of its precise data provides a framework for extraction of parton densities using the pseudodistributions approach. It was used in the recently performed calculations of the nucleon and pion valence quark distributions. We also discuss matching conditions for the pion distribution amplitude and generalized parton distributions, the lattice studies of which are now in progress.
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