  We consider discrete-time branching random walks with a radially symmetric
distribution. Independently of each other individuals generate offspring whose
relative locations are given by a copy of a radially symmetric point process
$\mathcal{L}$. The number of particles at time $t$ form a supercritical
Galton-Watson process. We investigate the maximal distance to the origin of
such branching random walks. Conditioned on survival, we show that, under some
assumptions on $\mathcal{L}$, it grows in the same way as for branching
Brownian motion or a broad class of one-dimensional branching random walks: the
first term is linear in time and the second logarithmic. The constants in front
of these terms are explicit and depend only on the mean measure of
$\mathcal{L}$ and dimension. Our main tool in the proof is a ballot theorem
with moving barrier which may be of independent interest.
