We define a generalization of contact geometry which is suitable for describing convex boundaries of symplectic manifolds which are non-exact at infinity. This generalizes the notion of weak symplectic fillings, in that the ``canonical'' structure at the boundary of a weak filling is this structure. This is also of interest to pure contact topology, in that the notion of a non-exact Legendrian is strictly larger than the class of Legendrians, even in a genuine contact manifold. This also gives rise to a notion of weak Lagrangian fillings of Legendrians, and a theory of non-exact Weinstein manifolds.