This work is a survey on the evolution of Operator Theory from its birth up its present state.Thus,after a brief sketch of the classical theory,we examine is some detail Grothendieck's fundamental work and then show how from this the theory developed to take its modern shape through the systematic use,largely due to Pietsch,of the concept of an operator ideal.Finally.the last section is dedicated to some general open problems which are fundamental for the theory.

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