The purpose of this paper is to establish, via a martingale approach, some refinements on the asymptotic behavior of the one-dimensional elephant random walk (ERW). The asymptotic behavior of the ERW mainly depends on a memory parameter p which lies between zero and one. This behavior is totally different in the diffusive regime 0 ≤ p < 3/4, the critical regime p = 3/4, and the superdiffusive regime 3/4 < p ≤ 1. Notwithstanding of this trichotomy, we provide some new results on the almost sure convergence and the asymptotic normality of the ERW.

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