Zimmer's superrigidity theorems on higher rank Lie groups and their lattices launched a program of study aiming to classify actions of semisimple Lie groups and their lattices, known as the {\it Zimmer program}. When the group is too large relative to the dimension of the phase space, the Zimmer conjecture predicts that the actions are all virtually trivial. At the other extreme, when the actions exhibit enough regular behavior, the actions should all be of algebraic origin.

I will present our recent progress in the program (will be self-contained). This is a joint work with D. Damjanovic, R. Spatzier and K. Vinhage , to appear in Acta Mathematica.