Consider the generalized flag manifold G/B and the corresponding affine flag manifold FlG. In this paper we use curve neighborhoods for Schubert varieties in FlG to construct certain affine Gromov-Witten invariants of FlG, and to obtain a family of "affine quantum Chevalley" operators Λ0,…,Λn indexed by the simple roots in the affine root system of G. These operators act on the cohomology ring H∗(FlG) with coefficients in Z[q0,…,qn]. By analyzing commutativity and invariance properties of these operators we deduce the existence of two quantum cohomology rings, which satisfy properties conjectured earlier by Guest and Otofuji for G=SLn(C). The first quantum ring is a deformation of the subalgebra of H∗(FlG) generated by divisors. The second ring, denoted QHaf∗(G/B), deforms the ordinary quantum cohomology ring QH∗(G/B) by adding an affine quantum parameter q0. We prove that QHaf∗(G/B) is a Frobenius algebra, and that the new quantum product determines a flat Dubrovin connection. Further, we develop an analogue of Givental and Kim formalism for this ring and we deduce a presentation of QHaf∗(G/B) by generators and relations. The ideal of relations is generated by the integrals of motion for the periodic Toda lattice associated to the dual of the extended Dynkin diagram of G.