I will describe two classes of gluing constructions for minimal surfaces in the 3-sphere; both generate sequences of minimal embeddings converging to singular configurations of multiple Clifford tori. One class, an extension of the Clifford torus doubling of Kapouleas and Yang (2010), produces minimal embeddings each of which resembles a stack of approximate Clifford tori connected by many small catenoidal tunnels arranged doubly periodically on the tori. The other class (joint work with Kapouleas) yields examples which resemble multiple Clifford tori intersecting along certain great circles, except that neighborhoods of the intersection circles have been replaced by approximate Karcher-Scherk towers so that the resulting surfaces are smoothly embedded. All these constructions proceed by first building a surface that is only approximately minimal but possesses the other desired properties and then finding a graph over this initial surface which is exactly minimal. This last step is accomplished by solving the relevant PDE, whose linearization has small eigenvalues that require special attention.