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.The O(N )-symmetric and Ising statistical models have played an extremely important role in our understanding of second order phase transitions both because many experimental systems show this symmetry and because they have been the playground on which almost all the theoretical formalisms aiming at describing these phase transitions have been developed and tested: Integrability 3 term becomes relevant in d = 3−ǫ and a nontrivial 2-unstable FP emerges from G that becomes 3-unstable. This scenario repeats in each critical dimension d n = 2 + 2/n below which a new n-unstable multicritical FP appears that we call T n . The FP T 2 is tricritical because it lies in the coupling constant space on the boarder separating the domain of second order and first order phase transitions. The common wisdom is that all the T n FPs can be followed by continuity in d down to d = 2 for all values of N . This is corroborated by the fact that in the Ising case (N = 1), it has been rigorously proven that indeed all the T n exist in d = 2 and are nontrivial [9]. Because of Mermin-Wagner theorem, the situation is physically different for N ≥ 2 but at least T 2 can be followed smoothly from d = 3 − ǫ down to d = 2 for N = 2, 3 and 4 [10]. Notice that the N = 2, d = 2 case is peculiar because topological defects can trigger in this case a finite-temperature phase transition.At N = ∞, exact results can be derived such as a closed and exact RG flow equation for the Gibbs effective potential [11]. The common wisdom is that at N = ∞ and in generic dimensions 2 < d < 4, the only nontrivial and nonsingular FP is WF which is simple to obtain after an appropriate rescaling by a factor N [12]. Its nonsingular character means that it is a regular function of the field. The limit N = ∞ is in fact peculiar because in all the d n with n ≥ 2, and only in these dimensions, there also exists a line of FPs. In d = 3, this line corresponds to tricritical FPs sharing all the same (trivial) critical exponents. This line starts at G and terminates at the Bardeen-Moshe-Bander (BMB) FP whose effective potential is nonanalytic at vanishing field, see .It is surprising that this common wisdom about the O(N ) models raises a simple paradox that, to the best of our knowledge, has remained unnoticed up to now.