We consider a nearest-neighbor random walk on Z, for which the probability of jumping along a bond of the lattice is proportional to exp[ - g. (number of previous jumps along that bond}~], with g > 0, h-E (0, 1]. After a review of earlier results obtained for the case r = l we outline the generalizations for k from (0, 1), obtaining a whole range of anomalous diffusion limits.