The theory of boundary-value problems for the Laplacian in Lipschitz domains is by now very well developed. Furthermore, many of the existing tools and known results for the Laplacian have been extended to the case of second-order linear equations of the form div (A grad u), where A is a matrix of variable coefficients. However, at present there are many open questions in the theory of higher-order elliptic differential equations. I will describe a generalization of layer potentials to the case of higher-order operators in divergence form; layer potentials are a common and very useful tool in the theory of second-order equations. I will then describe some applications of layer potentials to the theory of boundary-value problems. This is joint work with Steve Hofmann and Svitlana Mayboroda.