Under what conditions can we push-forward the differential forms on a manifold $M$ to differential forms on a manifold $N$ by a smooth map $f$?

Question

Let $M,N$ be two Complex Manifolds of the same (complex) dimension.

Let $f: M \longrightarrow N$ be a Smooth Map.

It is well-known that differential forms on the manifold $N$ can always be pulled-back to differential forms on the manifold $M$ by the map $f$.

Under what conditions can we push-forward the differential forms on the manifold $M$ to differential forms on the manifold $N$ by the map $f$?