This paper is devoted to the study of complex and unexpected phenomena that are observed in twodimensional mappings (or in three-dimensional flows) with homoclinic tangencies. In particular, we show that, in the C-topology, for an arbitrary finite r, in any neighborhood of a system with a quadratic homoclinic tangency there are nonrough systems with homoclinic tangencies of arbitrarily high orders, i.e., systems of an arbitrarily high codimension. Such phenomena were not observed in bifurcation theory previously. The study of systems with homoclinic tangencies was initiated in [4]. First and foremost, three classes of such systems were distinguished in that paper. Namely, let L be a saddle periodic motion, and let Γ be a homoclinic trajectory along which the stable and unstable invariant manifolds of L are quadratically tangent to each other (see Fig. 1). Let λ and γ be multipliers of L, |λ| 1. Assume that |λγ| = 1; moreover, without loss of generality, we can assume that |λγ| < 1. Let U be a small neighborhood of the closure Γ ∪ L of the homoclinic trajectory, and let N be the set of all trajectories that lie entirely in U . Depending on the signs of multipliers and on the signs of certain coefficients that characterize the way in which the stable and unstable manifolds adjoin to Γ, the systems with homoclinic tangencies fall into one of the following three classes: (1) for systems of the first class, the set N is trivial: N = {L,Γ}; (2) for systems of the second class, N is a nontrivial, nonuniformly hyperbolic set that admits a complete description in the language of symbolic dynamics (via some quotient system of the topological Bernoulli scheme consisting of three symbols); (3) for systems of the third class, N still contains nontrivial, hyperbolic subsets, but, generally speaking, the set N is not exhausted by them; moreover, the everywhere dense nonroughness takes place on bifurcation films of systems of the third class. (In [10, 11], a similar classification was carried out for the multidimensional case, including the case of systems with homoclinic tangencies of an arbitrary finite order.) To be more specific, according to [4], systems that have nonrough periodic motions are dense in any one-parameter family of systems with homoclinic tangencies of the third class in which the quantity