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The break-junction tunneling has been systematically investigated in MgB2. Two types of the break-junction contacts have been exploited on the same samples, which demonstrated tunnel contact like (SIS) and point contact like (SnS) behavior. Both of them have shown the existence of the two distinct energy gaps. We have found that the temperature dependence of a large gap Δ_σ(T) is qualitatively close to the BCS type but the ratio 2Δ_σ/kTc = 5.3 surpasses the BCS value: 3.5. The temperature dependence of a small gap Δ_π(T) can be different for different junctions and deviates significantly from the BCS type behavior. While investigating SnS contacts with different local Tc we have found that only a big 2D gap Δ_σ scaled with Tc. At the same time a small 3D gap Δ_π showed no reasonable tendency to change in the range 25 K ≤ Tc ≤ 40.5 K. We have observed also two-gap structures in the CVCs of stacks of SIS and SnS contacts, which proves a two-gap superconductivity to be an essentially intrinsic property of MgB2. Multi-band superconductors can possess some special collective excitations due to different values of the order parameter phases in different bands. The existence of such collective modes has been predicted by Leggett. We have clearly observed the manifestation of this mode with the energy ~4 meV in the CVCs of Josephson junctions and SnS-contacts at T < Tc.