**Critical exponents for O(n) models**

We consider the critical behaviour of long-range O(n) models for n greater than or equal to 0. For n=1,2,3,... these are phi^4 spin models. For n=0 it is the weakly self-avoiding walk. We prove existence of critical exponents for the susceptibility and the specific heat, below the upper critical dimension. This is a rigorous version of the epsilon expansion in physics. The proof is based on a rigorous renormalisation group method developed in previous work with Bauerschmidt and Brydges.