For natural numbers $k,n\geqslant 1$, let $\omega_k(n)$ be the number of prime factors of $n$ with multiplicity $k$. The functions $\omega_k(n)$ with $k\geqslant 1$ are refined versions of the well-known function $\omega(n)$ counting the number of distinct prime factors of $n$ without any conditions on the multiplicities. In this talk, we will cover several elementary, analytic and probabilistic results about the functions $\omega_k(n)$ with $k\geqslant 1$ and their function field analogues in polynomial rings with coefficients from a finite field. In particular, we will see that the function $\omega_1(n)$ and its function field analogue satisfy the Erdős-Kac Theorem. The results we will see in this talk are based on joint works with Yu-Ru Liu, with Sourabhashis Das, Wentang Kuo and Yu-Ru Liu, and with Greg Martin.