This chapter looks at the classes of semi-Fredholm operators in B(X) and attempts to classify them in terms of the mapping p. It shows that there is a connection between certain of the semi-Fredholm operators and the classes of right and left invertible elements in the quotient algebra B(X)/K(X). The chapter deals with two classic theorems about Fredholm operators. It presents the relationships between strictly singular and semi-Fredholm operators and between S(X) and other ideals in B(X). The reader is referred to I. Gohberg and M. Krein for an extended discussion and reference to sources. The chapter concludes by developing some specialized results.