In the infinite element method, the underlying domain is divided into infinitely many pieces. This leads to a system of infinitely many equations for infinitely many unknowns, but these can be reduced by analytical technicians to a finite system when some sort of scaling is present in the original problem. The simplest illustrative example, described carefully at the beginning of the first chapter of the book, is the solution of the Dirichlet problem in the exterior of some polygon. The exterior is subdivided into annular regions by a sequence of geometrically expanding images of the given polygon; these annuli are then further subdivided. The resulting variational equations take for the form of a block tridiagonal Toeplittz matrix, with an inhomogeneous term in the zero componenet. Various efficient methods are described for solving such systems of equations. The infinite element method is, wherever applicable, an elegant and efficient approach to solving problems in physics and engineering.