  A general notion of a quasi-finite algebra is introduced as an algebra graded
by the set of all integers equipped with topologies on the homogeneous
subspaces satisfying certain properties. An analogue of the regular bimodule is
introduced and various module categories over quasi-finite algebras are
described. When applied to the current algebras (universal enveloping algebras)
of vertex operator algebras satisfying Zhu's $C_2$-finiteness condition, our
general consideration derives important consequences on representation theory
of such vertex operator algebras. In particular, the category of modules over
such a vertex operator algebra is shown to be equivalent to the category of
modules over a finite-dimensional associative algebra.
