Let $E$ be a finite set in an abelian group, for instance a set of integers. Suppose we are given the second order correlation of this set, namely how many times each distance occurs between two points of $E$. Can we determine $E$? It is easy to see that $E$ cannot be determined from this information, not even up to translation and reflection (these operations obviously leave invariant the set of distances). What then if we are given the third order correlation, which is, for all $0<a<b$, for how many $x$ the numbers $x, x+a, x+b$ are all in $E$. Can we determine $E$ up to translation? In some cases the answer is yes but in general the answer is no. We are going to review this problem and some of its techniques which mix nicely combinatorial and number theoretic arguments as well as Fourier Analysis.

Joint work with Tamás Keleti and Philippe Jaming.