  The main aspects of a gauge-invariant approach to the description of quark
dynamics in the nonperturbative regime of QCD are first reviewed. In
particular, the role of the parallel transport operation in constructing
gauge-invariant Green's functions is presented, and the relevance of Wilson
loops for the representation of the interaction is emphasized. Recent
developments, based on the use of polygonal lines for the parallel transport
operation, are then presented. An integro-differential equation is obtained for
the quark Green's function defined with a phase factor along a single, straight
line segment. It is solved exactly and analytically in the case of
two-dimensional QCD in the large $N_c$ limit. The solution displays the
dynamical mass generation phenomenon for quarks, with an infinite number of
branch-cut singularities that are stronger than simple poles.
