We consider a variety of matrix models as a certain subspace of the whole space of integrable theories, namely, the subspace of forced ("semi-infinite") hierarchies. The integrability is discussed in terms of orthogonality conditions which generalize those of matrix models. The explicit solutions of matrix models are proposed using the fermionic representation of τ-functions. Various generalizations of matrix models associated with the generic point of the infinite flag space are introduced. The simplest example of a hermitian matrix model is investigated in detail and the first non-trivial example of unitary matrix model is also discussed. Finally, we point out that the cancellation of the partition function by positive Virasoro generators is a common thing for general τ-functions in integrable models and discuss the special role of the Virasoro algebra in matrix models, which can be interpreted as a gauge-fixing condition in the corresponding "string field theory".