We give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a free-monoid monad $T$ in our ambient category, and coequalisers satisfying some exactness conditions, we give an abstract envelope construction, putting paramonoids (and paracategories) in the more general context of partial algebras. We introduce for the latter the notion of saturation, which characterises those partial algebras which are isomorphic to the ones obtained from their enveloping algebras. We also set up a factorisation system for partial algebras, via inclusions and Kleene morphisms. We also explore some of the global aspects of the category of paracategories — (co)completeness and cartesian closure. We set-up the relevant notion of adjunction between paracategories, illustrated by the cartesian-closed paracategory of bivariant functors and dinatural transformations.