Suppose that K is a linear space of functions analytic in some domain D in the complex plane. A sequence Λ = (λk) of distinct points from D is said to be a set of uniqueness for K if ƒ∈K and ƒ(λk) = 0 for all k imply ƒ≡0. Depending on the dispersion and the density of Λ on the one hand, and the growth of the functions in K on the other, one may often require only |ƒ(λk)| [les ]ak for some sequence of positive numbers ak, and still conclude that ƒ≡0 for ƒ∈K. Of particular interest are sharp conditions on the decay of ak, which reflect the interplay between growth and decay of analytic functions.