Abstract : The range of viral epidemics, the amount of tissue affected by a propagating disease, or the volume of material affected by defect-spreading can all be thought as the volume explored by a branching random walk (BRW). Despite its wide range of applications, the scaling or growth rate of the volume explored by this process remains elusive, with exact results limited to one dimension. In this talk, I will present the exact calculation of the scaling and universal amplitudes of the volume explored by a BRW in the critical regime, the onset of epidemics. These results are valid on regular lattices of dimensions above and below the critical dimension dc=4, fractal lattices and complex networks. Our results characterise the spreading of spatial branching processes and provide a route to probe the spectral properties of social, metabolic and general complex networks.