  We develop further basic tools in the theory of continuous bounded cohomology
of locally compact groups. We apply this tools to establish a Milnor-Wood type
inequality in a very general context and to prove a global rigidity result
which was originally announced by the authors with a sketch of a proof using
bounded cohomology techniques and then proven by Koziarz and Maubon using
harmonic map techniques. As a corollary one obtains that a lattice in SU(p,1)
cannot be deformed nontrivially in SU(q,1), if either p is at least 2 or the
lattice is cocompact. This generalizes to noncocompact lattices a theorem of
Goldman and Millson.
