  We study the maximal displacement of a one dimensional subcritical branching
random walk initiated by a single particle at the origin. For each
$n\in\mathbb{N},$ let $M_{n}$ be the rightmost position reached by the
branching random walk up to generation $n$. Under the assumption that the
offspring distribution has a finite third moment and the jump distribution has
mean zero and a finite probability generating function, we show that there
exists $\rho>1$ such that the function \[ g(c,n):=\rho ^{cn} P(M_{n}\geq cn),
\quad \mbox{for each }c>0 \mbox{ and } n\in\mathbb{N}, \] satisfies the
following properties: there exist $0<\underline{\delta}\leq \overline{\delta} <
{\infty}$ such that if $c<\underline{\delta}$, then $$
0<\liminf_{n\rightarrow\infty} g (c,n)\leq \limsup_{n\rightarrow\infty} g (c,n)
{\leq 1}, $$ while if $c>\overline{\delta}$, then \[ \lim_{n\rightarrow\infty}
g (c,n)=0. \] Moreover, if the jump distribution has a finite right range $R$,
then $\overline{\delta} < R$. If furthermore the jump distribution is "nearly
right-continuous", then there exists $\kappa\in (0,1]$ such that
$\lim_{n\rightarrow \infty}g(c,n)=\kappa$ for all $c<\underline{\delta}$. We
also show that the tail distribution of $M:=\sup_{n\geq 0}M_{n}$, namely, the
rightmost position ever reached by the branching random walk, has a similar
exponential decay (without the cutoff at $\underline{\delta}$). Finally, by
duality, these results imply that the maximal displacement of supercritical
branching random walks conditional on extinction has a similar tail behavior.
