Operads describe a variety of algebraic structures and are particularly useful to organize hierarchies of higher homotopies. In some cases, it is required to relax the notion of operad by specifying domains of composable elements, which leads to the notion of partial operad. Two significant examples are considered: vertex operator algebras and partial operads on configuration spaces of points in a Riemannian manifold. Our approach is to define partial operads as paramonoids, exploring the fact that operads are monoids. Thus, we expect that the theory of paramonoids can be used to generalize operadic completion results, such as the Axelrod-Singer compactification. This talk is on ongoing joint work with C. Hermida.