It is by now generally accepted that the recently discovered superconductor MgB2 is a superconductor with two energy gaps. Here we discuss how this happens and address some of its unusual implications. The anisotropy ratio of the upper critical field in MgB2 single crystals shows an unusual strong temperature dependence. Measurements of the microwave conductivity on MgB2 thin films have shown an anomalous coherence peak appearing at significantly lower temperatures than in conventional superconductors. We argue that both of these observations are natural consequences of the existence of the two gaps, if the special Fermi surface structure of MgB2 is taken into account. We also examine consequences for the expected intrinsic nonlinear microwave response in MgB2. Calculations including impurity scattering indicate that under certain circumstances impurity scattering might improve the microwave properties of this material.