  We survey recent results of normal and anomalous diffusion of two types of
random motions with long memory in ${\Bbb R}^d$ or ${\Bbb Z}^d$. The first
class consists of random walks on ${\Bbb Z}^d$ in divergence-free random drift
field, modelling the motion of a particle suspended in time-stationary
incompressible turbulent flow. The second class consists of self-repelling
random diffusions, where the diffusing particle is pushed by the negative
gradient of its own occupation time measure towards regions less visited in the
past. We establish normal diffusion (with square-root-of-time scaling and
Gaussian limiting distribution) in three and more dimensions and typically
anomalously fast diffusion in low dimensions (typically, one and two). Results
are quoted from various papers published between 2012-2018, with some hints to
the main ideas of the proofs. No technical details are presented here.
