  We revisit the modular flavor symmetry from a more general perspective. The
scalar modular forms of principal congruence subgroups are extended to the
vector-valued modular forms, then we have more possible finite modular groups
including $\Gamma_N$ and $\Gamma'_N$ as the flavor symmetry. The theory of
vector-valued modular forms provide a method of differential equation to
construct the modular multiplets, and it also reveals the simple structure of
the modular invariant mass models. We review the theory of vector-valued
modular forms and give general results for the lower dimensional vector-valued
modular forms. The general finite modular groups are listed up to order 72. We
apply the formalism to construct two new lepton mass models based on the finite
modular groups $A_4\times Z_2$ and $GL(2,3)$.
