  We study the one-dimensional branching random walk in the case when the step
size distribution has a stretched exponential tail, and, in particular, no
finite exponential moments. The tail of the step size $X$ decays as
$\mathbb{P}[X \geq t] \sim a \exp(-\lambda t^r)$ for some constants $a, \lambda
> 0$ where $r \in (0,1)$. We give a detailed description of the asymptotic
behaviour of the position of the rightmost particle, proving almost-sure limit
theorems, convergence in law and some integral tests. The limit theorems reveal
interesting differences betweens the two regimes $ r \in (0, 2/3)$ and $ r \in
(2/3, 1)$, with yet different limits in the boundary case $r = 2/3$.
