  The rigidity propeties of higher rank diagonalizable actions is a major theme
in homogenous dynamics, with origins in work of Cassels and Swinnerton-Dyer in
the 1950s and Furstenberg. We survey both results and conjectures regarding
such actions, with emphasize on the applications of these results towards
understanding the distribution of integer points on varieties, quantum unique
ergodicity, and Diophantine approximations.
