I'm just beginning to learn (FIR) filter design and I was wondering, why is $\pi$ involved in the domain of such filters?

I assume it's related to some trigonometric functions, but why do I see e.g. cutoff frequency being related to $\pi$?

You can see for example here that the variable $\omega \in [-\pi, \pi]$ in function $z=e^{i \omega} $ of the transfer function of a FIR filter:

<https://en.wikipedia.org/wiki/Finite_impulse_response#Transfer_function>