(Joint with Henry Segerman.) It is a theorem of Moise that every three-manifold admits a triangulation, and thus infinitely many. Thus, it can be difficult to learn anything really interesting about the three-manifold from any given triangulation. Thurston introduced ``ideal triangulations'' for studying manifolds with torus boundary; Lackenby introduced ``taut ideal triangulations'' for studying the Thurston norm ball; Agol introduced ``veering triangulations'' for studying punctured surface bundles over the circle. Veering triangulations are very rigid; one current conjecture is that any fixed three-manifold admits only finitely many veering triangulations. After giving an overview of these ideas, we will introduce ``veering Dehn surgery''. We use this to give the first infinite families of veering triangulations with various interesting properties.