  Given a transformation $T$ of a standard measure space $(X,\mu)$, let $\Cal
M(T)$ denote the set of spectral multiplicities of the Koopman operator $U_T$
defined in $L^2(X,\mu)\ominus\Bbb C$ by $U_Tf:=f\circ T$. It is discussed in
this survey paper which subsets of $\Bbb N\cup\{\infty\}$ are realizable as
$\Cal M(T)$ for various $T$: ergodic, weakly mixing, mixing, Gaussian, Poisson,
ergodic infinite measure preserving, etc. The corresponding constructions are
considered in detail. Generalizations to actions of Abelian locally compact
second countable groups are also discussed.
