We determine several generalised Ramsey numbers for two sets Γ1 and Γ2 of cycles, in particular, all generalised Ramsey numbers R(Γ1, Γ2) such that Γ1 or Γ2 contains a cycle of length at most 6, or the shortest cycle in each set is even. This generalises previous results of Erdös, Faudree, Rosta, Rousseau, and Schelp from the 1970s. Notably, including both C3 and C4 in one of the sets, makes very little difference from including only C4. Furthermore, we give a conjecture for the general case. We also describe many (Γ1, Γ2)-avoiding graphs, including a complete characterisation of most (Γ1, Γ2)-critical graphs, i.e., (Γ1, Γ2)-avoiding graphs on R(Γ1, Γ2) − 1 vertices, such that Γ1 or Γ2 contains a cycle of length at most 5. For length 4, this is an easy extension of a recent result of Wu, Sun, and Radziszowski, in which |Γ1| = |Γ2| = 1. For lengths 3 and 5, our results are new even in this special case.