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Consider a mapping u from a smooth domain to a compact Riemannian manifold N whose Hessian energy \int | D2 u|^2 dx is finite. In joint work with T. Riviere, relations between the Hessian energy, the topological singularity of the map, and approximability by smooth maps is studied for W^{2,2}(B5,S3) and other topologically related cases. We also describe constructions needed for this work and for joint work with CY Wang on proving the optimal partial regularity of Hessian energy-minimizers with various target manifolds N.