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 N=4 and N=2 supersymmetric theories; many finite 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

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... e N=1 duality of Seiberg and duality in finite 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).