  We describe how and to what extent the noncommutative two-torus can be
approximated by a tower of finite-dimensional matrix geometries. The
approximation is carried out for both irrational and rational deformation
parameters by embedding the algebra of the noncommutative torus into an
approximately finite algebra. The construction is a rigorous derivation of the
recent discretizations of noncommutative gauge theories using finite
dimensional matrix models, and it shows precisely how the continuum limits of
these models must be taken. We clarify various aspects of Morita equivalence
using this formalism and describe some applications to noncommutative
Yang-Mills theory.
