  SrTiO$_3$ thin films were used as a model system to study the effects of
strain and epitaxial constraint on structural phase transitions of perovskite
films. The basic phenomena revealed will apply to a variety of important
structural transitions including the ferroelectric transition. Highly strained
SrTiO$_3$ films were grown on different substrates, providing both compressive
and tensile strain. The measured strain-temperature phase diagram is
qualitatively consistent with theory, however the increase in the phase
transition temperature is much larger than predicted. Due to the epitaxial
strain and substrate clamping, the SrTiO$_3$ lattice is tetragonal at all
temperatures. The phase transitions involve only changes in internal symmetry.
The low temperature phase under tensile strain has a unique structure with
orthorhombic $Cmcm$ space group but a tetragonal lattice, an interesting
consequence of epitaxial constraint.
