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There is a number of partial differential equations whose solutions are not scalar functions, but take their values in a nonlinear manifold.  Examples of such functions are configurations of liquid crystals and ferromagnets, elastic materials with orientation degrees of freedom, and fields of symmetric positive definite matrices.  The spaces of such functions are themselves nonlinear, which makes the discretization and numerical treatment of such partial differential equations notoriously difficult. We present generalizations of standard finite element methods to function spaces with a nonlinear target. These generalized FE spaces are H^1 conforming, and we show optimal a priori discretization error bounds.  The resulting algebraic problems are minimization problems on large products of low-dimensional manifolds, which can be solved efficiently by combining a monotone multigrid method with a Riemannian trust-region solver. We use the presented machinery to simulate solitons in ferromagnetic materials, and wrinkling patterns in thin elastic sheets, where we obtain good quantitative agreement with experimental results from the literature.