  We propose the use of algebras of generalized functions for the analysis of
certain highly singular problems in the calculus of variations. After a general
study of extremal problems on open subsets of Euclidean space in this setting
we introduce the first and second variation of a variational problem. We then
derive necessary (Euler-Lagrange equations) and sufficient conditions for
extremals. The concept of association is used to obtain connections to a
distributional description of singular variational problems. We study
variational symmetries and derive an appropriate version of N\"other's theorem.
Finally, a number of applications to geometry, mechanics, elastostatics and
elastodynamics are presented.
