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We find that the multicritical fixed point structure of the O($N$) models ismuch more complicated than widely believed. In particular, we find newnonperturbative fixed points in three dimensions ($d=3$) as well as at$N=\infty$. These fixed points come together with an intricate double-valuedstructure when they are considered as functions of $d$ and $N$. Many featuresfound for the O($N$) models are shared by the O($N)\otimes$O(2) models relevantto frustrated magnetic systems.

Surprises in the $O(N)$ models: nonperturbative fixed points, large $N$ limit and multi-criticality