We study various properties of quasimodular forms by using their connections
with Jacobi-like forms and pseudodifferential operators. Such connections are
made by identifying quasimodular forms for a discrete subgroup $\G$ of $SL(2,
\bR)$ with certain polynomials over the ring of holomorphic functions of the
Poincar\'e upper half plane that are $\G$-invariant. We consider a surjective
map from Jacobi-like forms to quasimodular forms and prove that it has a right
inverse, which may be regarded as a lifting from quasimodular forms to
Jacobi-like forms. We use such liftings to study Lie brackets and Rankin-Cohen
brackets for quasimodular forms. We also discuss Hecke operators and construct
Shimura isomorphisms and Shintani liftings for quasimodular forms.