  We obtain general trace formulae in the case of perturbation of self-adjoint
operators by self-adjoint operators of class $\bS_m$, where $m$ is a positive
integer. In \cite{PSS} a trace formula for operator Taylor polynomials was
obtained. This formula includes the Livshits--Krein trace formula in the case
$m=1$ and the Koplienko trace formula in the case $m=2$. We establish most
general trace formulae in the case of perturbation of Schatten--von Neumann
class $\bS_m$. We also improve the trace formula obtained in \cite{PSS} for
operator Taylor polynomials and prove it for arbitrary functions in he Besov
space $B_{\be1}^m(\R)$.
  We consider several other special cases of our general trace formulae. In
particular, we establish a trace formula for $m$th order operator differences.
