**On the critical branching random walk in supercritical and critical dimensions**

We extend several results in the potential theory of random walk to critical branching random walk. In the supercritical dimensions (\(d\geq 5\)), we introduce branching capacity for any finite subset of \(\mathbb{Z}^d\) and establish its connections with the hitting probability by branching random walk and branching recurrence. In the critical dimension (\(d=4\)), we also establish the asymptotics of the hitting probability and some related results.