  Nonequilibrium systems driven by additive or multiplicative dichotomous
Markov noise appear in a wide variety of physical and mathematical models. We
review here some prototypical examples, with an emphasis on {\em
analytically-solvable} situations. In particular, it has escaped attention till
recently that the standard results for the long-time properties of such systems
cannot be applied when unstable fixed points are crossed in the asymptotic
regime. We show how calculations have to be modified to deal with these cases
and present a few relevant applications -- the hypersensitive transport, the
rocking ratchet, and the stochastic Stokes' drift. These results reinforce the
impression that dichotomous noise can be put on a par with Gaussian white noise
as far as obtaining analytical results is concerned. They convincingly
illustrate the interplay between noise and nonlinearity in generating
nontrivial behaviors of nonequilibrium systems and point to various practical
applications.
