  We provide new general methods in the calculus of variations for the
anisotropic Plateau problem in arbitrary dimension and codimension. A new
direct proof of Almgren's 1968 existence result is presented; namely, we
produce from a class of competing "surfaces," which span a given bounding set
in some ambient space, one with minimal anisotropically weighted area. In
particular, rectifiability of a candidate minimizer is proved without the
assumption of quasiminimality. Our ambient spaces are a class of Lipschitz
neighborhood retracts which includes manifolds with boundary and manifolds with
certain singularities. Our competing surfaces are rectifiable sets which
satisfy any combination of general homological, cohomological or homotopical
spanning conditions. An axiomatic spanning criterion is also provided. Our
boundaries are permitted to be arbitrary closed subsets of the ambient space,
providing a good setting for surfaces with sliding boundaries.
