We define a rich model to be one which contains a proper elementary substructure isomorphic to itself. Existence, nonstructure, and categoricity theorems for rich models are proved. A theory $T$ which has fewer than $\min(2^\lambda,\beth_2)$ rich models of cardinality $\lambda(\lambda > |T|)$ is totally transcendental. We show that a countable theory with a unique rich model in some uncountable cardinal is categorical in $\aleph_1$ and also has a unique countable rich model. We also consider a stronger notion of richness, and use it to characterize superstable theories.