  We introduce the notion of tight homomorphism into a locally compact group
with nonvanishing bounded cohomology and study these homomorphisms in detail
when the target is a Lie group of Hermitian type. Tight homomorphisms between
Lie groups of Hermitian type give rise to tight totally geodesic maps of
Hermitian symmetric spaces. We show that tight maps behave in a functorial way
with respect to the Shilov boundary and use this to prove a general structure
theorem for tight homomorphisms. Furthermore we classify all tight embeddings
of the Poincare' disk.
