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We survey recent contributions to finite element exterior calculus onmanifolds and surfaces within a comprehensive formalism for the error analysisof vector-valued partial differential equations on manifolds. Our primary focusis on uniformly bounded commuting projections on manifolds: these projectionsmap from Sobolev de Rham complexes onto finite element de Rham complexes,commute with the differential operators, and satisfy uniform bounds in Lebesguenorms. They enable the Galerkin theory of Hilbert complexes for a large rangeof intrinsic finite element methods on manifolds. However, these intrinsicfinite element methods are generally not computable and thus primarily oftheoretical interest. This leads to our second point: estimating the geometricvariational crime incurred by transitioning to computable approximate problems.Lastly, our third point addresses how to estimate the approximation error ofthe intrinsic finite element method in terms of the mesh size. If the solutionis not continuous, then such an estimate is achieved via modified Cl\'ement orScott-Zhang interpolants that facilitate a broken Bramble--Hilbert lemma.

Towards finite element exterior calculus on manifolds: commuting projections, geometric variational crimes, and approximation errors