In recent years, Popa’s deformation/rigidity theory has lead to a wealth of classification and rigidity results for von Neumann algebras given by countable groups and their actions on measure spaces. In this talk, I will present the first rigidity and classification results for von Neumann algebras given by locally compact groups and their actions. I establish that the crossed product von Neumann algebra has a unique Cartan subalgebra, when the action is free and probability measure preserving and the group is a connected simple Lie group of real rank one, or a group acting properly on a tree. From this, I deduce a W*-strong rigidity result for irreducible actions of products of such groups. I also establish that the group von Neumann algebra of such groups are strongly solid. More generally, our results hold for locally compact groups that are non-amenable, weakly amenable and belong to Ozawa’s class S. This is joint work with Arnaud Brothier and Stefaan Vaes.