We study the coherent properties of transmission through Kondo impurities, by considering an open Aharonov-Bohm ring with an embedded quantum dot. We develop a novel many-body scattering theory which enables us to calculate the conductance through the dot, the transmission phase shift, and the normalized visibility, in terms of the single-particle T-matrix. For the single-channel Kondo effect, we find at temperatures much below the Kondo temperature T_K that the transmission phase shift is π/2

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... ithout any corrections up to order (T/T_K)^2. The visibility has the form 1-(π T/T_K)^2. For the non-Fermi liquid fixed point of the two channel Kondo, we find that transmission phase shift is π/2 despite the fact that a scattering phase shift is not defined. At zero temperature the visibility is 1/2, thus at zero temperature exactly half of the conductance is carried by single-particle processes. We explain that the spin summation masks the inherent scattering phases of the dot, which can be accessed only via a spin-resolved experiment. In addition, we calculate the effect of magnetic field and channel anisotropy, and generalize to the k-channel Kondo case.