To start with, some basic facts about Lyapunov exponents first for linear and then for nonlinear (deterministic as well as random) dynamical systems are recalled/introduced. Then invariant measures for random dynamical systems (RDS) are discussed in more detail. For RDS with independent increments (e.g. given by products of independent maps or by stochastic differential equations) two notions are available here. The more classical one comes from the associated Markov semigroup, while the RDS notion is a
pathwise one. There is a rather close relation between these two.
Finally, a characterization of invariant measures with a non-positive leading Lyapunov exponent is given.