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In this article, we study the maximal displacement in a branching randomwalk. We prove that its asymptotic behaviour consists in a first almost sureballistic term, a negative logarithmic correction in probability andstochastically bounded fluctuations. This result, proved by Addario-Berry andReed, and Hu and Shi is given here under close-to-optimal integrabilityconditions. We provide simple proofs for this result, also deducing thegenealogical structure of the individuals that are close to the maximaldisplacement.

Asymptotic of the maximal displacement in a branching random walk