  Nodal lines, as one-dimensional band degeneracies in momentum space, usually
feature a linear energy splitting. Here, we propose the concept of magnetic
higher-order nodal lines, which are nodal lines with higher-order energy
splitting and realized in magnetic systems with broken time reversal symmetry.
We provide sufficient symmetry conditions for stabilizing magnetic quadratic
and cubic nodal lines, based on which concrete lattice models are constructed
to demonstrate their existence. Unlike its counterpart in nonmagnetic systems,
the magnetic quadratic nodal line can exist as the only band degeneracy at the
Fermi level. We show that these nodal lines can be accompanied by torus surface
states, which form a surface band that span over the whole surface Brillouin
zone. Under symmetry breaking, these magnetic nodal lines can be transformed
into a variety of interesting topological states, such as three-dimensional
quantum anomalous Hall insulator, multiple linear nodal lines, and magnetic
triple-Weyl semimetal. The three-dimensional quantum anomalous Hall insulator
features a Hall conductivity $\sigma_{xy}$ quantized in unit of $e^2/(hd)$
where $d$ is the lattice constant normal to the $x$-$y$ plane. Our work reveals
previously unknown topological states, and offers guidance to search for them
in realistic material systems.
