The minisymposium focuses on recent progress in a new area of mathematical physics and applied analysis, namely, on spectral theory and differential equations defined of metric graphs. Studies of physical systems which are either made low-dimensional or discrete for the purposes of computations or inherently built on networks are rapidly growing. The subject of differential equations on metric graphs (or "quantum graphs") which model waveguides, nanotubes, and many other devices is the area where the tools of spectral theory, dynamical systems, and analysis of PDEs have to be adapted to the new setting of increased geometric complexity. Some recent progress in this area will be presented by mostly junior mathematicians specializing on linear and nonlinear theory of quantum graphs. Speakers of the minisymposium will cover various aspects of eigenvalue estimates for the Laplace operator on graphs, localization of eigenfunctions of the Schrodinger equation (both linear and nonlinear), nodal domains, quantum resonances, and existence and stability of nonlinear waves on graphs.