  We survey recent results in hermitian integral geometry, i.e. integral
geometry on complex vector spaces and complex space forms. We study valuations
and curvature measures on complex space forms and describe how the global and
local kinematic formulas on such spaces were recently obtained. While the local
and global kinematic formulas in the Euclidean case are formally identical, the
local formulas in the hermitian case contain strictly more information than the
global ones. Even if one is only interested in the flat hermitian case, i.e.
$\mathbb C^n$, it is necessary to study the family of all complex space forms,
indexed by the holomorphic curvature $4\lambda$, and the dependence of the
formulas on the parameter $\lambda$. We will also describe Wannerer's recent
proof of local additive kinematic formulas for unitarily invariant area
measures.

Differential Geometry