Maps on matrix spaces

It is well known that every automorphism of the full matrix algebra is inner. We give a short proof of this statement and discuss several extensions of this theorem including structural results for multiplicative maps on matrix algebras, characterizations of monotone and orthogonality preserving maps on idempotent matrices, some nonlinear preserver results, and some recent theorems concerning geometry of matrices. We show that all these topics are closely related and point out the connections with physics and geometry. Several open problems are posed.