These are lecture notes from a course given at the summer school "Heat kernels and spectral geometry: from manifolds to graphs" in Bregenz, Austria, 2022. They are designed to be accessible to doctoral level students, and include background chapters on Laplacians on domains and quantum graphs before moving on to specialised topics involving the dependence and optimisation of operator eigenvalues on a metric graph in function of the graph geometry, drawn in part from the recent literature.

Comment: Final version, to appear in Communications in Mathematics