In terms of the algebraic language introduced by S. Alesker, the integral geometry of the euclidean group acting on R power n may be summarized in the statement: the algebra of euclidean- invariant convex valuations on R power n is isomorphic to the truncated polynomial algebra R[t]/(t power n+1). In this talk we present an extensive picture of the integral geometry of the unitary group acting on C power n, which is much more complicated and interesting. This is joint work with Andreas Bernig.