A discrete group is said to be inner amenable if it admits an atomless mean which is invariant under conjugation. In this talk I will provide a satisfying characterization of inner amenability for linear groups over an arbitrary field. I will also discuss a complete characterization of linear groups which are stable in the sense of Jones and Schmidt. The analysis of stability leads to many new examples of (non-linear) stable groups; notably, all nontrivial countable subgroups of Monod's group H(R) are stable. This includes nonamenable groups constructed by Monod and by Lodha and Moore, as well as Thompson's group F.