The first part of this monograph is an elementary introduction to the theory of Frechet algebras. Important examples of Frechet algebras, which are among those considered, are the algebra of all holomorphic functions on a (hemicompact) reduced complex space, and the algebra of all continuous functions on a suitable topological space. The problem of finding analytic structure in the spectrum of a Frechet algebra is the subject of the second part of the book. In particular, the author pays attention to function algebraic characterizations of certain Stein algebras (= algebras of holomorphic functions on Stein spaces) within the class of Frechet algebras.