  We study the so-called elephant random walk (ERW) which is a non-Markovian
discrete-time random walk on $\mathbb{Z}$ with unbounded memory which exhibits
a phase transition from diffusive to superdiffusive behaviour. We prove a law
of large numbers and a central limit theorem. Remarkably the central limit
theorem applies not only to the diffusive regime but also to the phase
transition point which is superdiffusive. Inside the superdiffusive regime the
ERW converges to a non-degenerate random variable which is not normal. We also
obtain explicit expressions for the correlations of increments of the ERW.
