We propose a series of conjectures concerning the Hilbert functions of points (or more generally zero-dimensional subschemes) in projective space. We begin by extending the results of Castelnuovo and others on points in uniform position, and then consider the corresponding problem without the hypothesis of uniform position. A special case is a conjectured extension of the ical Cayley-Bacharach theorem. We prove this conjecture in projective space $\mathbb {P}^r$ for all $r \leq 7$. Finally we make a conjecture extending Macaulay's theorem on the Hilbert function of graded rings, and discuss its relation to the previous conjectures.