  By applying the recently developed nonperturbative functional renormalization
group (FRG) approach, we study the interplay between ferromagnetism, quasi-long
range order (QLRO) and criticality in the $d$-dimensional random field O(N)
model in the whole ($N$, $d$) diagram. Even though the "dimensional reduction"
property breaks down below some critical line, the topology of the phase
diagram is found similar to that of the pure O(N) model, with however no
equivalent of the Kosterlitz-Thouless transition. In addition, we obtain that
QLRO, namely a topologically ordered "Bragg glass" phase, is absent in the
3--dimensional random field XY model. The nonperturbative results are
supplemented by a perturbative FRG analysis to two loops around $d=4$.
