In this talk we show that any II$_1$ factor associated to a free ergodic probability measure preserving action $\Gamma\curvearrowright X$ of a product $\Gamma=\Gamma_1\times\dots\times\Gamma_n$ of icc hyperbolic, free product or wreath product groups is prime (i.e. $L^\infty(X)\rtimes\Gamma$ cannot be decomposed as a tensor product of II$_1$ factors), provided $\Gamma_i\curvearrowright X$ is ergodic, for any $1\leq i\leq n$. We also present a unique prime factorization result for any II$_1$ factor arising from a free ergodic probability measure preserving action of a product of icc, hyperbolic, property (T) groups.