  The classical algebra $\Lambda$ of symmetric functions has a remarkable
deformation $\Lambda^*$, which we call the algebra of shifted symmetric
functions. In the latter algebra, there is a distinguished basis formed by
shifted Schur functions $s^*_\mu$, where $\mu$ ranges over the set of all
partitions. The main significance of the shifted Schur functions is that they
determine a natural basis in $Z(\frak{gl}(n))$, the center of the universal
enveloping algebra $U(\frak{gl}(n))$, $n=1,2,\ldots$.
  The functions $s^*_\mu$ are closely related to the factorial Schur functions
introduced by Biedenharn and Louck and further studied by Macdonald and other
authors.
  A part of our results about the functions $s^*_\mu$ has natural classical
analogues (combinatorial presentation, generating series, Jacobi--Trudi
identity, Pieri formula). Other results are of different nature (connection
with the binomial formula for characters of $GL(n)$, an explicit expression for
the dimension of skew shapes $\lambda/\mu$, Capelli--type identities, a
characterization of the functions $s^*_\mu$ by their vanishing properties,
`coherence property', special symmetrization map $S(\frak{gl}(n))\to
U(\frak{gl}(n))$.
  The main application that we have in mind is the asymptotic character theory
for the unitary groups $U(n)$ and symmetric groups $S(n)$ as $n\to\infty$.
  The results of this paper were used in \cite{Ok1--3}.
