  We call a random point measure infinitely ramified if for every $n\in \mathbb
N$, it has the same distribution as the $n$-th generation of some branching
random walk. On the other hand, branching L\'evy processes model the evolution
of a population in continuous time, such that individuals move in space
independently, according to some L\'evy process, and further beget progenies
according to some Poissonian dynamics, possibly on an everywhere dense set of
times. Our main result connects these two classes of processes much in the same
way as in the case of infinitely divisible distributions and L\'evy processes:
the value at time $1$ of a branching L\'evy process is an infinitely ramified
point measure, and conversely, any infinitely ramified point measure can be
obtained as the value at time $1$ of some branching L\'evy process.
