  We develop a theory of ergodicity for a class of random dynamical systems
where the driving noise is not white. The two main tools of our analysis are
the strong Feller property and topological irreducibility, introduced in this
work for a class of non-Markovian systems. They allow us to obtain a criteria
for ergodicity which is similar in nature to the Doob--Khas'minskii theorem.
The second part of this article shows how it is possible to apply these results
to the case of stochastic differential equations driven by fractional Brownian
motion. It follows that under a nondegeneracy condition on the noise, such
equations admit a unique adapted stationary solution.
