This thesis discusses regularity problems of minimizing maps and flows for various functionals and targets. It consists of four parts: Part 1. Energy Minimizing Mappings into Polyhedra. We prove both the partial interior and complete boundary regularities for maps which minimize energy among all maps into a polyhedron. Part 2. Bubbling Phenomena of Certain Palais-Smale Sequences from Surfaces into General Targets. We show that there is no unaccounted loss of energy for certain Palais-Smale sequences from a surface into a general manifold during the process of bubbling. We also discuss the harmonicity of weak limits of general Palais-Smale sequences. Part 3. Maps Minimizing Convex Functionals between Riemannian Manifolds. We show that any map, which minimizes a uniformly strictly convex C\sp2 functionals F among all maps from one manifold to another manifold, has Holder continuous first gradient away from a closed subset with Lebesgue measure zero. Part 4. Existence and Partial Regularity of Weak Flows of Convex Functionals. Assume we are given a C\sp2 convex functional F, we prove the existence of a weak flow associated to it. We also prove that such a weak flow has Holder continuous spatial gradient away from a closed subset with Lebesgue measure zero.