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4 pages, 4 figures.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 $d=4$

Unified Picture of Ferromagnetism, Quasi-Long-Range Order, and Criticality in Random-Field Models