How does the pushforward act on a differential form?

Question

Let $F:M\mapsto N$ be a smooth map of smooth manifolds and $\omega\in\Omega^k(M)$. In the lecture notes that I'm working with, I see the notation $F_*\omega$. I'm assuming that this is the push forward of $F$ acting on $\omega$ and I loosely understand what this map does, but I don't understand how one would define it rigorously and can't find anything about it online. Some help would be appreciated.