Abstract: Many invariants of knots and links in $S^3$ can be understood in terms of spanning surfaces---compact surfaces embedded in $S^3$ with boundary equal to a given knot or link. Murasugi sum, also called generalized plumbing, is a particular way of gluing together two spanning surfaces along a disk. I will begin by describing how these gluings work and how they improve our understanding of at least some of the following: the Alexander polynomial, the genus of a knot, fiberedness, incompressibility and the fundamental group, Tait's flyping conjecture, Kauffman states, state surfaces, Khovanov homology, and Heegaard Floer homology. Next, I will explain how plumbing unknotted annuli and Mobius bands in the pattern of a tree yields several important classes of links: 2-bridge, pretzel, Montesinos, and arborescent. The 2-bridge case is particularly rich, as described in a beautiful paper by Hatcher and Thurston, and I may go into some detail about current work on this topic. Finally, I will describe other current work that adapts Murasugi sum to an operation on spanning solids in $S^4$. Expect lots of pictures!