"A generalized geometric framework for constrained systems" A geometric framework for constrained dynamical systems is presented. It allows to describe in a unified way a general type of first order singular differential equations on a manifold; these equations can not be written in normal form since the derivatives appear multiplied by a *linear* operator, therefore we call them *linearly constrained systems*. The concepts of *constraints* and *morphisms* between lin- early constrained systems are defined, and their relationships studied. Finally, a *stabilization algorithm* is devised and carefully discussed in order to solve the equation of motion. Our formalism includes the presymplectic and the lagrangian formalisms, as well as higher order lagrangians, and we give several applications of it; in particular, a stabilization algorithm for the lagrangian formalism is obtained.