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