In the classical simple random walk the steps are independent, viz., the walker has no memory. In contrast, in the elephant random walk which was introduced by Schütz and Trimper in 2004, the walker remembers the whole past, and the next step always depends on the whole path so far. Our main aim is to prove analogous results when the elephant has only a restricted memory, for example remembering only the most remote step(s), the most recent step(s) or both. We also extend the models to cover more general step sizes.