A recent topic including a joint work with Bertrand Rémy will be shown. We will deal with not only the simplicity of some Kac-Moody group, but also its universal covering. Recall that a universal Kac-Moody group is obtained by the so-called Tits group functor associated with a certain realization of a given generalized Cartan matrix. Furthermore, we would like to discuss the corresponding Schur multiplier.