Morse theory offers a way to look at smooth manifolds as being built up out of particularly simple pieces, called handles. I will start by describing this process and relating it to the notions of surgery and cobordism. Next, I will introduce Morse homology, which is a sort of “smooth cellular homology” (where cells are replaced by handles) with powerful applications. Morse homology provides a blueprint, by which we can manipulate the handle presentation of a manifold into a particularly simple form (at least in higher dimensions). This leads into the h-cobordism theorem, which was a shocking revelation in differential topology, showing for the first time that the structure of manifolds is sometimes easier to understand when the dimension is large. In particular, I will describe how this theorem helps us identify uniquely defining properties of balls and spheres.