This expository talk will be dealing with infinite-dimensional Lie theory as defined independently by R. Moody and V. Kac. One of the goals of the talk is to stress some peculiarities of the objects obtained by starting from a generalized Cartan matrix which is neither of spherical nor of affine type. It is well-known that the relevant Weyl groups are then very different from the more classical ones (the latter groups are finite or virtually abelian). This difference is reflected even sharply in some more elaborate objects such as Kac-Moody groups. These groups were defined very early by R. Moody and K. Teo in their initial version; J. Tits provided later some elaborations leading to a vast generalization of Chevalley group schemes. Such groups, and completions of them, have surprising properties at least from the viewpoint of discrete group theory and harmonic analysis.