  We review recent developments in the Casimir effect which arises in
quantization volumes restricted by material boundaries and in spaces with
non-Euclidean topology. The starting point of our discussion is the novel exact
solution for the electromagnetic Casimir force in the configuration of a
cylinder above a plate. The related work for the scalar Casimir effect in
sphere-plate configuration is also considered, and the application region of
the proximity force theorem is discussed. Next we consider new experiments on
the measurement of the Casimir force between metals and between metal and
semiconductor. The complicated problem connected with the theory of the thermal
Casimir force between real metals is analyzed in detail. The present situation
regarding different theoretical approaches to the resolution of this problem is
summarized. We conclude with new constraints on non-Newtonian gravity obtained
using the results of latest Casimir force measurements and compare them with
constraints following from the most recent gravitational experiments.
