There are many physically interesting superconformal gauge theories in four dimensions. In this talk I discuss a common phenomenon in these theories: the existence of continuous families of infrared fixed points. Well-known examples include finite ${cal N}=4$ and ${cal N}=2$ supersymmetric theories; many finite ${cal N}=1$ examples are known also. These theories are a subset of a much larger class, whose existence can easily be established and understood using the algebraic methods explained here. A relation between the ${cal N}=1$ duality of Seiberg and duality in finite ${cal N}=2$ theories is found using this approach, giving further evidence for the former. This talk is based on work with Robert Leigh (hep-th/9503121).
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