Hodge theory is a standard theory in characterizing differential complexes and the topology of manifolds. This text is a study of the Hodge-Kodaira and related decompositions on manifolds with boundary under mainly analytic aspects and aims at developing a method for solving boundary value problems. Analyzing a Dirichlet form on the exterior algebra bundle allows to give a refined version of the classical decompostion results of Morrey. A projection technique leads to existence and regularity theorems for a wide class of boundary vlaue problems for differential forms and vector fields. The book links aspects of the geometry of manifolds with the theory of partial differential equations. It is intended to be comprehensible for graduate students and mathematicians working in either of these fields.