Stochastic processes driven by Gaussian yet power-law correlated noise, suchas Mandelbrot's fractional Brownian motion (FBM) are ubiquitous in a range ofcomplex systems, e.g., for the motion of tracers in viscoelastic environments,in "rough" financial data, or for the persistent motion of animals. FBM is anergodic yet strongly non-Markovian process, with often surprising behaviour.In this talk I will briefly introduce these processes and demonstrate that instrong confinement their probability density may assume multimodal shapes, while in soft external potentials no steady state exists. An application tobrain fibre growth is discussed. In heterogeneous environments the dynamicsof a diffusing test particle may become a (random or deterministic) functionof time. For these cases I will introduce novel extensions of FBM such asmemory-multimodal FBM and FBM with a "diffusing diffusivity". Finally, I willdiscuss tempering of the power-law correlations, to emulate systems with afinite correlation time. This description is a good description for observeddynamics of lipid molecules in model membranes..