  Branching processes are widely used to model phenomena from networks to
neuronal avalanching. In a large class of continuous-time branching processes,
we study the temporal scaling of the moments of the instant population size,
the survival probability, expected avalanche duration, the so-called avalanche
shape, the $n$-point correlation function and the probability density function
of the total avalanche size. Previous studies have shown universality in
certain observables of branching processes using probabilistic arguments,
however, a comprehensive description is lacking. We derive the field theory
that describes the process and demonstrate how to use it to calculate the
relevant observables and their scaling to leading order in time, revealing the
universality of the moments of the population size. Our results explain why the
first and second moment of the offspring distribution are sufficient to fully
characterise the process in the vicinity of criticality, regardless of the
underlying offspring distribution. This finding implies that branching
processes are universal. We illustrate our analytical results with computer
simulations.
