Various philosophical accounts of function attributions have taken the following form: f is a function of a structure X in a system S if and only if X does f in S and f causally contributes to G. While sharing this form, these accounts disagree over how “G” is to be specified. Specifications of “G” range from the fairly determinate (e.g., “the reproductive capacity of S”, or “the survival of S”) to the less determinate (e.g., “some goal of S” or “some good of S”). Although much of the debate over functions has been concerned with the proper characterization of “G”, it has become apparent that theories which fit this schema are subject to now-standard counterexamples in which f does in fact causally contribute to the realization of G, but f fails to be a function of X in that f's contribution is accidental in nature. Consider, for example, goal-state theory according to which f is a function of X in S just in case f contributes to the realization of some goal of S. Imagine that Jones has a silver dollar in his shirt pocket which deflects a bullet meant for his chest. Assuming that the coin is in Jones's pocket for reasons unrelated to his goal of continued existence, although the coin contributes to this goal, it does not have the function of “bullet deflector” since its contribution is merely accidental. We can easily generate similar counter-examples for other theories which fit this schema but differ in how “G” is specified. I will call this the problem of “accidental G-generating effects”. Counterexamples of this type have put pressure on theorists to modify this schema or at least to add additional conditions in order to rule out “accidental functions”. In this paper I will first examine various modifications which have been suggested for avoiding this type of counterexample, and then develop a new approach to the question of “accidental functions”.