We consider geodesic flows of Riemannian metrics on T 2 and S 2 integrable by an integral depending on the momenta linearly or quadratically (in what follows such integrals are referred to as linear and quadratic integrals) and show that there exists an effective criterion for topological equivalence of two arbitrary flows. The criterion is based on comparing the "codes" of the corresponding Riemannian metrics (see below). A. T. Fomenko [1, 2] found a new topological invariant I(H, Q) of integrable Hamiltonian systems, making it possible to classify integrable Hamiltonians up to the coarse topological equivalence. In this case the labeled topological invariant I*(H, Q) discovered by A. T. Fomenko and H. Wsishang [3] (also called the labeled molecule [4]) classifies integrable Hamiltonians up to a finer topological equivalence. Let M 4 be a smooth symplectic manifold, and let v = sgrad H be an integrable Hamiltonian system with smooth Hamiltonian H . Following [1] we say that an integral F on an isoenergetic surface Q3 = {H = const} is a Bott integral if all its critical points form nondegenerate critical submanifolds. Definition (see [4]). A pair (P1, K1), where P1 is a compact orientable surface and K1 a nonempty finite connected graph on P1, is called a letter-atom if the following conditions hold: 1) the degree of each vertex of 1(1 is equal to 0, 2, or 4; 2) each connected component of the difference P1 \ K1 is homeomorphic to the annulus S 1 x (0, 1]; 3) the set of the annuli forming P1 \ :K~ can be divided into two parts (positive and negative annuli) so that exactly one positive and one negative annulus adjoin each edge of K1. A molecule W, which is a union of atoms, is a pair ( P , K ) , where K is a graph embedded in a two-dimensional surface p2. A labeled molecule W* is obtained by supplementing W with numerical (rational or integer) labels. To each unlabeled molecule W we assign its complexity (m, n) , where m is the number of vertices in K (coinciding with the total number of critical circles of the Bott integral) and n is the number of connected components of P \ K (or the number of cylinders S 1 x S 1 x (0, 1) into which the isoenergetic manifold Q3 splits On deleting all singular leaves of the corresponding Liouville foliation). Fomenko posed the problem of describing all labeled molecules W* for the above type of integrable geodesic flows on T 2 and S 2 and of finding the corresponding (m, n)-domain in the molecular complexity table. For the torus T 2 this problem has been completely investigated by Selivanova whose results, as it turned out later, can be applied effectively to the topological classification of geodesic flows on the sphere S 2 with nontrivial quadratic integral. Here we also consider geodesic flows on S 2 with linear integral and thus complete the investigation of integrable geodesic flows on two-dimensional Riemannian manifolds (with additional integral of degree no higher than 2). As is known [6, 9], the geodesic flows on two-dimensional surfaces of genus > 1 have no additional integrals analytic with respect to the momenta. In §2 we give the basic facts concerning the above type of geodesic flows that are used in the subsequent presentation and are based on the results of Darboux [10], Levi-Civita [11], Birkhoff [5], and Kolokol'tsov