  We consider the volumes of classical supermanifolds such as the supersphere,
complex projective superspace, and Stiefel and Grassmann supermanifolds, with
respect to the natural metrics or symplectic structures. We show that the
formulas for the volumes, upon certain universal normalization, can be obtained
by an analytic continuation from the formulas for the volumes of the
corresponding ordinary manifolds.
  Volumes of nontrivial supermanifolds may identically vanish. In 1970s,
Berezin discovered that the total Haar measure of the unitary supergroup
$\un(n|m)$ vanishes unless $m=0$ or $n=0$, i.e., unless it reduces to the
ordinary unitary group $\un(n)$ or $\un(m)$. Witten recently suggested that the
(Liouville) volume of a compact even symplectic supermanifold should always be
zero if it is not an ordinary manifold. Our calculations provide
counterexamples to this conjecture. On the other hand, we give a simple
explanation of Berezin's statement and generalize it to the Stiefel
supermanifold $\st_{r|s}(\C{n|m})$. There are also possible connections with
the recent works by Mkrtchyan and Veselov on `universal formulas' in Lie
algebra theory.
