  We establish some limit theorems for one-dimensional elephant random walk,
including Berry-Esseen bounds, Cram\'{e}r moderate deviations and local limit
theorems. These limit theorems can be regarded as refinements of the central
limit theorems for the elephant random walk. Moreover, by these limit theorems,
we conclude that the domain of attraction of normal distribution mainly depends
on a memory parameter $p$ which lies between $0$ and $3/4.$
