The orbit structure of finite measure preserving actions of non-compact Lie groups exhibits a wide range of very strong rigidity properties. The main theorem of the paper describes those countable groups with finite measure preserving actions that are stably orbit equivalent to such an action of a higher rank simple Lie group. This is applied to obtain information on the question of when an ergodic equivalence relation is generated by a free action of a group. Another application concerns foliations and actions with a transverse geometric structure. The precise statements of the results are too technical to be given here in detail.