  In this note we study four dimensional theories with N=3 superconformal
symmetry, that do not also have N=4 supersymmetry. No examples of such theories
are known, but their existence is also not ruled out. We analyze several
properties that such theories must have. We show that their conformal anomalies
obey a=c. Using the N=3 superconformal algebra, we show that they do not have
any exactly marginal deformations preserving N=3 supersymmetry, or global
symmetries (except for their R-symmetries). Finally, we analyze the possible
dimensions of chiral operators labeling their moduli space.
