In this thesis I discuss the application of two topological structures to scientific visualization. The first is the contour tree, a structure that represents the connectivity of the level sets of a function. I describe methods to compute the contour tree from piecewise-quadratic functions, and I develop a volume rendering framework that uses the contour tree to apply individual transfer functions to topologically distinct regions of the dataset. The second structure I consider is the separating surface of a multiphase 3D segmentation, i.e., a segmentation containing many more than two regions. Specifically, I consider the problem of constructing this separating surface given a series of 2D cross-sections. Two methods are described: A numerical method that operates on a voxel grid and produces smooth triangulated surface using a nearly-minimal number of triangles, and a topological method that operates only on the combinatorial structure of the segmentation and produces a cell complex that connects prescribed regions in adjacent cross-sections. For both the contour tree and the separating surface, properties of the described methods and algorithms are proved, implementation details are discussed, and experimental results are presented.