  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).
