Calkin algebras were first studied seriously by Calkin in the special but very interesting case where X is a separable Hilbert space. The treatment of Calkin algebras over Banach spaces was begun by Yood. This chapter provides a modernized version of what is referred to as the "Riesz-Schauder theory." Their results have been considerably generalized and the original arguments are obsolete. The original paper by F. Riesz, although now fifty-six years old, can still be warmly recommended to all readers. This pioneering paper is unusual for its elegance and charm. Extensions of some of the results from the case of bounded linear operators to that of closed unbounded linear operators is important for applications and of considerable intrinsic interest.