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In this note we study four dimensional theories with N=3 superconformalsymmetry, that do not also have N=4 supersymmetry. No examples of such theoriesare known, but their existence is also not ruled out. We analyze severalproperties that such theories must have. We show that their conformal anomaliesobey a=c. Using the N=3 superconformal algebra, we show that they do not haveany exactly marginal deformations preserving N=3 supersymmetry, or globalsymmetries (except for their R-symmetries). Finally, we analyze the possibledimensions of chiral operators labeling their moduli space.

On four dimensional N=3 superconformal theories