  If $\rho$ denotes a finite dimensional complex representation of
$\textbf{SL}_2(\textbf{Z})$, then it is known that the module $M(\rho)$ of
vector valued modular forms for $\rho$ is free and of finite rank over the ring
$M$ of scalar modular forms of level one. This paper initiates a general study
of the structure of $M(\rho)$. Among our results are absolute upper and lower
bounds, depending only on the dimension of $\rho$, on the weights of generators
for $M(\rho)$, as well as upper bounds on the multiplicities of weights of
generators of $M(\rho)$. We provide evidence, both computational and
theoretical, that a stronger three-term multiplicity bound might hold. An
important step in establishing the multiplicity bounds is to show that there
exists a free-basis for $M(\rho)$ in which the matrix of the modular derivative
operator does not contain any copies of the Eisenstein series $E_6$ of weight
six.
