The subject of this thesis is lens design using a system of functional differential equations arising from Fermat's Principle in geometrical optics. The emphasis is primarily on existence, uniqueness, and analyticity, properties of solutions to these equations, but some asymptotic methods are developed for special cases. Three specific lens problems are considered in detail: the first is an axial lens having two pairs of foci on the optical axis, the second is an axial lens which focuses light at two different frequencies to two distinct points, the third is a lens symmetric about an axis having foci not on said axis.