  We give both sufficient conditions and necessary conditions for the
stochastic stability of non-uniformly expanding maps either with or without
critical sets. We also show that the number of probability measures describing
the statistical asymptotic behaviour of random orbits is bounded by the number
of SRB measures if the noise level is small enough.
  As an application of these results we prove the stochastic stability of
certain classes of non-uniformly expanding maps introduced in \cite{V} and
\cite{ABV}.
