Mixed Logical Dynamical (MLD) systems are introduced as a new system type. The MLD form is capable to model a broad class of systems arising in many applications: linear hybrid systems; sequential logical systems (finite state machines, automata); nonlinear dynamic systems, where the nonlinearity can be expressed through combinational logic; some classes of discrete event systems; constrained linear systems. Controllability/verification and observability of MLD systems and other system theoretic properties are defined. Tests for these properties are formulated in the form of Mixed-Integer Linear Programs. Moving horizon control and estimation strategies with stability guarantees are proposed. These strategies require the iterative solution of Mixed-Integer Quadratic Programs. Several examples communicate the power and versatility of the proposed framework.