We are interested in the maximum of supercritical branching random walk or branching Brownian motion in the real line, which under some mild condition, shifted by $a*n-b\log n$, converges in law to some non-degenerate distribution. We first study the lower and moderate deviation probabilities for this convergence and obtain different regimes in Schröder case and Bëtcher case. Next, we consider the process conditioned on atypically small maximum and describe its behaviours. This is based on the joint works with Hui He and Bastien Mallein.