In deformation theory, higher structure naturally occurs on two different levels. Firstly, it occurs on the complexes governing deformations: according to Deligne’s principle, these should at least be dg Lie or L infinity-algebras. Secondly, higher structure can occur when modeling deformed objects: these may be more complicated than the original objects in case the deformation complex encodes additional structure. Taking Gerstenhaber’s deformation theory of algebras, governed by the Hochschild complex, as the starting point, we will discuss the occurrence of higher structure when deforming schemes into “noncommutative spaces”. We will look into two algebraic models for spaces: dg categories and prestacks. For the latter, we will present an operadic structure on the Gerstenhaber-Schack complex recently established in joint work with Hoang Dinh Van and Lander Hermans. This yields an underlying L infinity-structure governing deformations of prestacks.