  Many graphene moir\'e superlattices host narrow bands with non-zero valley
Chern numbers. We provide analytical and numerical evidence for a robust spin
and/or valley polarized insulator at total integer band filling in nearly flat
bands of several different moir\'e materials. In the limit of a perfectly flat
band, we present analytical arguments in favor of the ferromagnetic state
substantiated by numerical calculations. Further, we numerically evaluate its
stability for a finite bandwidth. We provide exact diagonalization results for
models appropriate for ABC trilayer graphene aligned with hBN, twisted double
bilayer graphene, and twisted bilayer graphene aligned with hBN. We also
provide DMRG results for a honeycomb lattice with a quasi-flat band and
non-zero Chern number, which extend our results to larger system sizes. We find
a maximally spin and valley polarized insulator at all integer fillings when
the band is sufficiently flat. We also show that interactions may induce
effective dispersive terms strong enough to destabilize this state. These
results still hold in the case of zero valley Chern number (for example,
trivial side of TLG/hBN). We give an intuitive picture based on extended
Wannier orbitals, and emphasize the role of the quantum geometry of the band,
whose microscopic details may enhance or weaken ferromagnetism in moir\'e
materials.
