As a generalization of the notion of measure, valuations on convex bodies have long played a central role in geometry. The starting point for many important new results in valuation theory is Hadwiger's remarkable characterization of the continuous rigid motion invariant real valued valuations as linear combinations of the intrinsic volumes. Among many applications, this result allows an effortless proof of the famous Principal Kinematic Formula from integral geometry.

In the first part of this talk, the decomposition of the space of continuous translation invariant valuations into a sum of SO(n) irreducible subspaces is presented. It will be explained how this result can be reformulated in terms of a Hadwiger type theorem for translation invariant and SO(n) equivariant valuations with values in an arbitrary (finite dimensional) SO(n) module. From this perspective the classical theorem of Hadwiger becomes the special case when the SO(n) module is the trivial 1-dimensional one.

A striking recent development in valuation theory explores the connections between isoperimetric inequalities and convex body valued valuations. To be more specific, many powerful geometric inequalities involve fundamental operators on convex bodies which are valuations, e.g. projection and intersection body maps. In many instances the proofs of these inequalities are based on the symmetry of certain bivaluations associated with convex body valued valuations.

In the second part, the decomposition of the space of translation invariant valuations into irreducible SO(n) modules is used to study the symmetry of O(n) invariant bivaluations and to establish new Brunn-Minkowski type inequalities for convex body valued valuations.