This paper analyzes the role of sharing rules in the existence and form of Bertrand equilibria for general demand and cost functions. It derives characteristics of equilibria that either do not depend on sharing rules or only on very general properties. We show that sign-preserving sharing leads to existence and uniqueness of zero-profit equilibria under fairly mild assumptions, while all other rules, including the ``classical rule'' of equal sharing, may either lead to non-existence of pure and mixed equilibria, or existence of a continuum of positive-profit equilibria.