We find that the multicritical fixed point structure of the O(N) models is much more complicated than widely believed. In particular, we find new nonperturbative fixed points in three dimensions (d=3) as well as at N=∞. These fixed points come together with an intricate double-valued structure when they are considered as functions of d and N. Many features found for the O(N) models are shared by the O(N)⊗O(2) models relevant to frustrated magnetic systems.