  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
$N=\infty$. 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($N)\otimes$O(2) models relevant
to frustrated magnetic systems.
