I will first review `normal' deterministic diffusion in a simple
piecewise linear chaotic map, where the mean square displacement of an
ensemble of particles grows linearly in time. Analytical results yield
a diffusion coefficient that is a fractal function of a control
parameter, a phenomenon that should be seen in experiments [1]. As a
second example, I will discuss diffusion in the nonlinear climbing sine
map. The bifurcation scenario exhibited by this map leads to a
complicated scenario between normal and anomalous diffusion, where the
mean square displacement grows nonlinearly in time [2], depending on
control parameters. Finally I will study the Pomeau-Manneville map,
which displays intermittency by exhibiting weak chaos. Crosslinks to
the new mathematical field of infinite ergodic theory will be briefly
outlined. Stochastic continuous time random walk theory predicts
subdiffusion for this map, that is, anomalous diffusion with a mean
square displacement that grows less than linearly in time [2,3], as is
confirmed by computer simulations. In a scaling limit a fractional
diffusion equation will be derived. However, as in case of normal
diffusion there are complicated fractal, possibly discontinuous
parameter dependencies deviating from the predictions of stochastic
theory.