Random Matrix Theory arose as an attempt to understand, in a qualitative way, the behavior of large quantum mechanical systems in the setting of randomness and symmetry. Subsequent developments brought into play remarkably effective methods from quantum field theory to analyze the asymptotic behavior of these mechanical systems that in turn led to surprising connections with classical dynamical problems. This talk will present an elementary survey of some of these connections, in particular Laplace growth models, nonlinear conservation laws and topological transitions between symmetric and Hamiltonian systems.