  We study the non-perturbative renormalization group flow of the nonlinear
O(N) sigma model in two and three spacetime dimensions using a scheme that
combines an effective local Hybrid Monte Carlo update routine, blockspin
transformations and a Monte Carlo demon method. In two dimensions our results
verify perturbative renormalizability. In three dimensions, we determine the
flow diagram of the theory for various $N$ and different truncations and find a
non-trivial fixed point, which indicates non-perturbative renormalizability. It
is related to the well-studied phase transition of the O(N) universality class
and characterizes the continuum physics of the model. We compare the obtained
renormalization group flows with recent investigations by means of the
Functional Renormalization Group.
