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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.
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