We present detailed theoretical and experimental analysis of the magneto-optic transverse Kerr effect in magnetic multilayers. The theoretical model is based upon a phenomenological permittivity tensor. From the general result, suitable only for numerical calculations, we derive several approximate analytical expressions in order to make a qualitative discussion. The theoretical predictions are compared with experimental results in Y/Co bilayers, and the good agreement found allows for an accurate determination of the magneto-optical constants of the material. Then, the theoretical model is applied to make a detailed study of interface magnetism in ${\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{Co}}_{x}$ alloys, and to perform numerical simulations in Co/Cu and Fe/Cu multilayers. The results in multilayers highlight the complex behavior of the magneto-optic transverse Kerr effect, in which the contributions of the individual layers are never strictly additive. This nonlinearity is found to be strongly dependent on the $3d$ magnetic metal present and could be used to probe the alignment of the layers even in a configuration of vanishing magnetic moment.