  We present a short overview of the recent results in the theory of diffusion
and wave equations with generalised derivative operators. We give generic
examples of such generalised diffusion and wave equations, which include
time-fractional, distributed order, and tempered time-fractional diffusion and
wave equations. Such equations exhibit multi-scaling time behaviour, which
makes them suitable for the description of different diffusive regimes and
characteristic crossover dynamics in complex systems.
