abstract: Random dynamical systems (RDS) provide useful and flexible models to investigate systems evolving under noisy or externally forced mechanisms. In the first part of the talk, we illustrate how transfer operators and multiplicative ergodic theory may be combined to shed light on fundamental dynamical information about RDS, including invariant measures, exponential decay rates and coherent structures, which characterize dominant transport aspects in these systems. Then, we present examples of random maps where such features, encoded in the so-called Lyapunov–Oseledets spectrum, can be understood and analyzed under perturbations.