In the first part of the talk, we give an overview of various ways in which deformation theory shapes conceptions of noncommutative spaces. In the second part of the talk, we focus on prestacks which occur as deformations of schemes. We introduce the framework of box operads in order to establish the relevant higher structure on the Gerstenhaber-Schack complex of a prestack, allowing for a self-contained treatment of the higher order deformation theory of prestacks. This is joint work with Hoang Dinh Van and Lander Hermans.