Abstract: Two different classes of random walks (and diffusions) with long memory will be considered. (A) Self-repelling walks (and diffusions) where the long term memory is caused by the tracer particle being pushed away from locations highly visited in the past. (B) Walks (and diffusions) in static random environment where the local random drift field is divergence-free. For cases in class (A) super-diffusive scaling is established in 1 and 2 dimensions and normal diffusion in 3 and more. For cases in class (B) sharp conditions are established for the normal diffusive behaviour and some interesting low-dimensional examples where super-diffusive scaling is either proved or expected to hold. The talk will be a survey of various results from the last decade and open problems for the next one.