Let \(A:\Omega\to \mathbb{R}^{n^2}\) be a measurable matrix function in an open set \(\Omega\subset \mathbb{R}^n\). The authors are concerned with the \(A\)-harmonic operator \(\text{\textsterling} u:= \text{div}(A\nabla u)\) acting on the Sobolev space \(W^{1,p}_0(\Omega)\). It is assumed that \textsterling{} is uniformly elliptic and that the entries of \(A\) are of vanishing mean oscillation. The main theorem says that the map \(\text{\textsterling}:W^{1,p}(\mathbb{R}^{n})\to W^{-1,p}(\mathbb{R}^n)\) is invertible for any \(p\in(1,\infty)\). The proof is based on recent estimates for the Riesz transforms combined with Fredholm index theory. An extensive introduction presents basic ideas and many interesting comments. Section 2 relates \(A\)-harmonic functions and quasiregular mappings. Section 3 deals with an integral form of the \(A\)-harmonic equation. Section 4 gives a proof of the main theorem.