  We establish a correspondence between vector-valued modular forms with
respect to a symmetric tensor representation and quasimodular forms. This is
carried out by first obtaining an explicit isomorphism between the space of
vector-valued modular forms with respect to a symmetric tensor representation
and the space of finite sequences of modular forms of certain type. This
isomorphism uses Rankin-Cohen brackets and extends a result of Kuga and
Shimura, who considered the case of vector-valued modular forms of weight two.
We also obtain a correspondence between such vector-valued modular forms and
weak Jacobi forms.
