The properties of the distribution of resonance zones in the two-planet problem are analyzed as a function of the semi-major axes and masses of the planets on cosmogonic time scales. A solution to equations averaged using the Hori—DePrit method with accuracy to second order in the small parameter of the problem is used, and the translation from the averaged to the osculating elements taken into account. Conditions for the overlap of resonance zones are obtained. It is shown that motion of pairs of planets in the extrasolar planetary systems HD 73526, 47 UMa, HD 181433 (c-d), GJ 581 (b-c), and HD 155358 can occur in regions of resonance overlap. The orbital evolution of the two-planet system 47 UMa is analyzed in the absence of resonance, in the vicinity of the 1: 2, 3: 7, 2: 5, 3: 8,1: 3, and 1: 7 resonances when the initial data correspond to a region of wide resonance, and in regions of overlap of wide-resonance zones. The possibility of chaotic motions of the planets in the 47 UMa system is demonstrated.