  The honeycomb Kitaev model describes a $Z_2$ spin liquid with topological
order and fractionalized excitations consisting of gapped $\pi$-fluxes and free
Majorana fermions. Competing interactions, even when not very strong, are known
to destabilize the Kitaev spin liquid. Magnetic fields are a convenient
parameter for tuning between different phases of the Kitaev systems, and have
even been investigated for potentially counteracting the effects of other
destabilizing interactions leading to a revival of the topological phase. Here
we review the progress in understanding the effects of magnetic fields on some
of the perturbed Kitaev systems, particularly on fractionalization and
topological order.
