Kinematic formulas on real vector spaces were shown by
Chern-Blaschke-Santaló in the 1930s. It was a long standing open problem
to find the corresponding formulas on complex vector spaces, or on
complex projective and complex hyperbolic spaces. In recent joint works
with Joseph Fu and Gil Solanes, we used a blending of geometric and
algebraic methods to solve this problem. I will give a survey on these
techniques and state some of the main results.

[invited by S. Wenger]