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