We consider control and inverse problems on metric graphs for several types of PDEs including the wave, heat and Schr\"odinger equations. We demonstrate that, for graphs without cycles, unknown coefficients of the equations together with the topology of the graph and lengths of the edges can be recovered from the dynamical Dirichlet-to-Neumann map associated to the boundary vertices. For general graphs with cycles additional observations at the internal vertices are needed for stable identification. The corresponding exact controllability results are also proved.