  We present a detailed study of the geometric and algebraic properties of the
multidimensional quadrilateral lattice (a lattice whose elementary
quadrilaterals are planar; the discrete analogue of a conjugate net) and of its
basic reductions. To make this study, we introduce the notions of forward and
backward data, which allow us to give a geometric meaning to the tau-function
of the lattice, defined as the potential connecting these data. Together with
the known circular lattice (a lattice whose elementary quadrilaterals can be
inscribed in circles; the discrete analogue of an orthogonal conjugate net) we
introduce and study two other basic reductions of the quadrilateral lattice:
the symmetric lattice, for which the forward and backward data coincide, and
the D-invariant lattice, characterized by the invariance of a certain natural
frame along the main diagonal. We finally discuss the Egorov lattice, which is,
at the same time, symmetric, circular and D-invariant. The integrability
properties of all these lattices are established using geometric, algebraic and
analytic means; in particular we present a D-bar formalism to construct large
classes of such lattices. We also discuss quadrilateral hyperplane lattices and
the interplay between quadrilateral point and hyperplane lattices in all the
above reductions.
