  We present a systematic calculation of the volumes of compact manifolds which
appear in physics: spheres, projective spaces, group manifolds and generalized
flag manifolds. In each case we state what we believe is the most natural scale
or normalization of the manifold, that is, the generalization of the unit
radius condition for spheres. For this aim we first describe the manifold with
some parameters, set up a metric, which induces a volume element, and perform
the integration for the adequate range of the parameters; in most cases our
manifolds will be either spheres or (twisted) products of spheres, or quotients
of spheres (homogeneous spaces).
  Our results should be useful in several physical instances, as instanton
calculations, propagators in curved spaces, sigma models, geometric scattering
in homogeneous manifolds, density matrices for entangled states, etc. Some flag
manifolds have also appeared recently as exceptional holonomy manifolds; the
volumes of compact Einstein manifolds appear in String theory.
