Polly is a high-level loop and data-locality optimizer and optimization
infrastructure for LLVM. It uses an abstract mathematical representation based
on integer polyhedra to analyze and optimize the memory access pattern of a
program. We currently perform classical loop transformations, especially tiling
and loop fusion to improve data-locality. Polly can also exploit OpenMP level
parallelism, expose SIMDization opportunities. Work has also be done in the
area of automatic GPU code generation.

For many users, however, it's not the existing optimizations in Polly that are
of most interest, but the new analyses and optimizations enabled by the Polly
infrastructure. At polyhedral.info you can get an idea of what has already been
done and what is possible in the context of polyhedral compilation.