## ABSTRACT

There are several interactions between the general theory of commutative topological algebras over C with unity and that of the holomorphic functions of one or several complex variables. This chapter divides in two parts: first will be reviewed known results on the general theory of Banach algebras, Frechet algebras, LB and LF algebras, as well as the open problems in the field. The second part will be examined concrete algebras of holomorphic functions, again reviewing known results and open problems. The chapter focuses on the interplay between the two theories, showing how each one can be considered either an object of study per se or a mean to understand better the other subject. The Gelfand topology is the weakest topology on the spectrum M(B) that makes every Gelfand transform a continuous map. Finiteness properties of a topological algebra and structure properties of its spectrum have interesting links.