In terms of the algebraic language introduced by S. Alesker, the integral geometry
 of the euclidean group acting on R&#94;n may be summarized in the statement: the
 algebra of euclidean- invariant convex valuations on R&#94;n is isomorphic to the
    truncated polynomial algebra R(t)/(t&#94;{n+1}). In this talk we present an
extensive picture of the integral geometry of the unitary group acting on C&#94;n,
  which is much more complicated and interesting. This is joint work with Andreas
                                      Bernig.