  A functional renormalization group approach to $d$-dimensional,
$N$-component, non-collinear magnets is performed using various truncations of
the effective action relevant to study their long distance behavior. With help
of these truncations we study the existence of a stable fixed point for
dimensions between $d= 2.8$ and $d=4$ for various values of $N$ focusing on the
critical value $N_c(d)$ that, for a given dimension $d$, separates a first
order region for $N<N_c(d)$ from a second order region for $N>N_c(d)$. Our
approach concludes to the absence of stable fixed point in the physical -
$N=2,3$ and $d=3$ - cases, in agreement with $\epsilon=4-d$-expansion and in
contradiction with previous perturbative approaches performed at fixed
dimension and with recent approaches based on conformal bootstrap program.
