A general system of noncoaxial separated astigmatic optical elements is examined. Allowance is made for prismatic elements including decentered lenses and refracting surfaces, prisms, and plane interfaces. The directions and positions of a ray entering and emerging from the system are related by a system matrix and a system vector. Given an incident ray one can readily obtain the emergent ray. The analysis is of fundamental importance for visual optics. Two numerical examples are presented. One is a model eye with astigmatic and obliquely crossing astigmatic and decentered surfaces.