Following Dyson, we treat the eigenvalues of a random matrix as a system of particles undergoing random walks. The dynamics of large matrices is then well described by fluid dynamical equations. In particular, the inviscid Burgers’ equation is ubiquitous and controls the behavior of the spectral density of large matrices. The solutions to this equation exhibit shocks that we interpret as the edges of the spectrum of eigenvalues. Going beyond the large N limit, we show that the average characteristic polynomial (or the average of the inverse characteristic polynomial) obeys equations that are equivalent to a viscid Burgers’ equation, or equivalently a diffusion equation, with 1∕N playing the role of the viscosity and encoding the entire finite N effects. This approach allows us to recover in an elementary way many results concerning the universal behavior of random matrix theories and to look at QCD spectral features from a new perspective.