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In this paper, we discuss two well-known open problems in the regularitytheory for nonlinear, conformally invariant elliptic systems in dimensions$n\ge 3$, with a critical nonlinearity: $H$-systems (equations of hypersurfacesof prescribed mean curvature) and $n$-harmonic maps into compact Riemannianmanifolds.  For $n=2$ several solutions of these problems are known but they all breakdown in higher dimensions (unless one considers special cases, e.g.hypersurfaces of constant mean curvature or manifolds with symmetries). Wediscuss some of the known proofs and hint at the main difficulties.  We also state a few new results (such as positive answers for all solutionsof class $W^{n/2,2}$ for even $n$, instead of $W^{1,n}$) and list some openquestions of independent interest - including specific endpoint variants of theCoifman-Rochberg-Weiss theorem, addressing the boundedness of commutators offractional and singular integrals with multiplication by bounded functions ofclass $W^{1,n}$ - that would lead to solutions of these two problems.

Invitation to H-systems in higher dimensions: known results, new facts, and related open problems