We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter τ \tau having positive imaginary part. When τ → i ∞ \tau \to i\infty , these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable τ \tau . We obtain a reciprocity law for these sums, which gives rise to new relations between certain modular forms (of one variable).