  The first half of this dissertation reviews the basic notion of vector-valued
modular forms and its connection to differential equations. The main purpose of
the dissertation is to classify spaces of vector-valued modular forms
associated to irreducible, T-unitarizable representations of the full modular
group, of dimension less than six. Given such a representation, it is shown
that the associated graded complex linear space of vector-valued modular forms
is a free module over the ring of integral weight modular forms for the full
modular group, whose rank is equal to the dimension of the given
representation. An explicit basis is computed for the module structure in each
case, and this basis is used to compute the Hilbert-Poincare series associated
to each graded space.
