Abstract A wide range of problems arising in practical applications can be formulated as Mixed-Integer Nonlinear Programs (MINLPs). For the case in which the objective and constraint functions are convex, some quite effective exact and heuristic algorithms are available. When non-convexities are present, however, things become much more difficult, since then even the continuous relaxation is a global optimization problem. We survey the literature on non-convex MINLPs, discussing applications, algorithms, and software. Special attention is paid to the case in which the objective and constraint functions are quadratic.