We present the operator semigroups approach to the first- and second-order dynamical systems taking place on metric graphs. We briefly survey the existing results and focus on the well-posedness of the problems with standard vertex conditions. Finally, we show two applications to biological models. This article is part of the theme issue 'Semigroup applications everywhere'.
Keywords: diffusion equation; genetic mutation model; networks; operator semigroups; synaptic transmission model; transport equation.