  Random diffeomorphisms with bounded absolutely continuous noise are known to
possess a finite number of stationary measures. We discuss dependence of
stationary measures on an auxiliary parameter, thus describing bifurcations of
families of random diffeomorphisms. A bifurcation theory is developed under
mild regularity assumptions on the diffeomorphisms and the noise distribution
(e.g. smooth diffeomorphisms with uniformly distributed additive noise are
included). We distinguish bifurcations where the density function of a
stationary measure varies discontinuously or where the support of a stationary
measure varies discontinuously.
  We establish that generic random diffeomorphisms are stable. Densities of
stable stationary measures are shown to be smooth and to depend smoothly on an
auxiliary parameter, except at bifurcation values. The bifurcation theory
explains the occurrence of transients and intermittency as the main bifurcation
phenomena in random diffeomorphisms. Quantitative descriptions by means of
average escape times from sets as functions of the parameter are provided.
Further quantitative properties are described through the speed of decay of
correlations as function of the parameter.
  Random endomorphisms are studied in one dimension; we show that stable one
dimensional random endomorphisms occur open and dense and that in one parameter
families bifurcations are typically isolated. We classify codimension one
bifurcations for one dimensional random endomorphisms; we distinguish three
possible kinds, the random saddle node, the random homoclinic and the random
boundary bifurcation. The theory is illustrated on families of random circle
diffeomorphisms and random unimodal maps.
