  Given a super-critical branching random walk on $\mathbb R$ started from the
origin, let $M_n$ be the maximal position of individuals at the $n$-th
generation. Under some mild conditions, it is known from \cite{A13} that as
$n\rightarrow\infty$, $M_n-x^*n+\frac{3}{2\theta^*}\log n$ converges in law for
some suitable constants $x^*$ and $\theta^*$. In this work, we investigate its
moderate deviation, in other words, the convergence rates of
$$\mathbb{P}\left(M_n\leq x^*n-\frac{3}{2\theta^*}\log n-\ell_n\right),$$ for
any positive sequence $(\ell_n)$ such that $\ell_n=O(n)$ and
$\ell_n\uparrow\infty$. As a by-product, we also obtain lower deviation of
$M_n$; i.e., the convergence rate of
  \[
  \mathbb{P}(M_n\leq xn),
  \] for $x<x^*$ in B\"{o}ttcher case where the offspring number is at least
two. Finally, we apply our techniques to study the small ball probability of
limit of derivative martingale.
