Friedrich' theory of symmetric positive systems of first-order PDE's is revisited so as to avoid invoking traces at the boundary. Two intrinsic geometric conditions are introduced to characterize admissible boundary conditions. It is shown that the space in which admissible boundary conditions can be enforced is maximal in a positive cone associated with the differential operator. The equivalence with a formalism based on boundary operators is investigated and practical means to construct these boundary operators are presented. Finally, the link with Friedrich' formalism and applications to various PDE's are discussed.