The authors consider the role of mirrors in synchrotron-radiation beamlines and discuss the optical considerations involved in their design. They discuss toroidal, spherical, elliptical, and paraboloidal mirrors in detail with particular attention to their aberration properties. They give a treatment of the sine condition and describe its role in correcting the coma of axisymmetric systems. They show in detail how coma is inevitable in single-reflection, grazing-incidence systems but correctable in two-reflection systems such as those of the Wolter type. In an appendix, they give the theory of point aberrations of reflectors of a general shape and discuss the question of correct naming of aberrations. In particular, a strict definition of coma is required if attempts at correction are to be based on the sine condition.