  Let $\Lambda$ be a basic finite dimensional algebra over an algebraically
closed field, presented as a path algebra modulo relations; further, assume
that $\Lambda$ is graded by lengths of paths. The paper addresses the
classifiability, via moduli spaces, of classes of graded $\Lambda$-modules with
fixed dimension $d$ and fixed top $T$. It is shown that such moduli spaces
exist far more frequently than they do for ungraded modules. In the local case
(i.e., when $T$ is simple), the graded $d$-dimensional $\Lambda$-modules with
top $T$ always possess a fine moduli space which classifies these modules up to
graded-isomorphism; moreover, this moduli space is a projective variety with a
distinguished affine cover that can be constructed from quiver and relations of
$\Lambda$. When $T$ is not simple, existence of a coarse moduli space for the
graded $d$-dimensional $\Lambda$-modules with top $T$ forces these modules to
be direct sums of local modules; under the latter condition, a finite
collection of isomorphism invariants of the modules in question yields a
partition into subclasses, each of which has a fine moduli space (again
projective) parametrizing the corresponding graded-isomorphism classes.
