I give a short review of the recent results obtained on
stochastic processes with space and time dependent
diffusivities. In the first part of the talk the
properties of heterogeneous diffusion process with
space-dependent diffusivity of a power-law form
D(x)~|x|^\alpha are considered. In the second part we
discuss the properties of scaled Brownian motion with
time-dependent diffusivity D(t)~t^\alpha. For both
processes we study correlation properties, non-ergodic
behavior and behavior in a confinement.