Triangulations appear everywhere, from the volume computations and meshing to algebra and topology. This book studies subdivisions and triangulations of polyhedral regions and point sets, and presents the first comprehensive treatment of the theory of secondary polytopes and related topics. A central theme of the book is the use of the rich structure of the space of triangulations to solve computational problems (e.g., counting the number of triangulations or finding optimal triangulations with respect to various criteria), as well as for setting up connections to applications in algebra, computer science, combinatorics, and optimization. With many examples and exercises, and with almost five hundred illustrations, the book gently guides beginners through the properties of the spaces of triangulations of "structured" (e.g., cubes, cyclic polytopes, lattice polytopes) and "pathological" (e.g., disconnected spaces of triangulations) situations using only elementary principles.