In a joint work with D. Hoff and A. Ioana, we have discovered the following product rigidity phenomenon: if $\Gamma$ is an icc group measure equivalent to a product of non-elementary hyperbolic groups, then any tensor product decomposition of the II$_1$ factor $L(\Gamma)$ arises only from the canonical direct product decomposition of $\Gamma$. Subsequently, I. Chifan, R. de Santiago and W. Sucpikarnin classified all the tensor product decompositions for group von Neumann algebras arising from a large class of amalgamated free products. In this talk we will give an overview of these results and discuss about a similar rigidity phenomenon that appears in the context of von Neumann algebras arising from actions. More precisely, we prove that if $\Gamma$ is a product of certain groups and $\Gamma\curvearrowright (X,\mu)$ is an arbitrary free ergodic measure preserving action, then we show that any tensor product decomposition of the II$_1$ factor $L^\infty(X)\rtimes\Gamma$ arises only from the canonical direct product decomposition of the underlying action $\Gamma\curvearrowright X.$