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We study the extremes of branching random walks under the assumption that theunderlying Galton-Watson tree has infinite progeny mean. It is assumed that thedisplacements are either regularly varying or they have lighter tails. In theregularly varying case, it is shown that the point process sequence ofnormalized extremes converges to a Poisson random measure. We study theasymptotics of the scaled position of the rightmost particle in the $n$-thgeneration when the tail of the displacement behaves like $\exp(-K(x))$, whereeither $K$ is a regularly varying function of index $r> 0$, or $K$ has anexponential growth. We identify the exact scaling of the maxima in all casesand show the existence of a non-trivial limit when $r> 1$.

Branching random walk with infinite progeny mean: a tale of two tails