  We present a novel approach to quantifying the morphology of Cosmic Microwave
Background (CMB) anisotropy maps. As morphological descriptors, we use shape
parameters known as Minkowski functionals. Using the mathematical framework
provided by the theory of integral geometry on arbitrary curved supports, we
point out the differences to their characterization and interpretation in the
case of flat space. With restrictions of real data -- such as pixelization and
incomplete sky coverage, to mention just a few -- in mind, we derive and test
unbiased estimators for all Minkowski functionals. Various examples, among them
the analysis of the four-year COBE DMR data, illustrate the application of our
method.
