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 $NN_c(d)$. Our approach concludes to the absence of stable fixed point in the physical - $N=2,3$ and $d=3$ - cases, in agreement with $��=4-d$-expansion and in contradiction with previous perturbative approaches performed at fixed dimension and with recent approaches based on conformal bootstrap program.

16 pages, 8 figures