  In electronic band structures, nodal lines may arise when two (or more) bands
contact and form a one-dimensional manifold of degeneracy in the Brillouin
zone. Around a nodal line, the dispersion for the energy difference between the
bands is typically linear in any plane transverse to the line. Here, we perform
an exhaustive search over all 230 space groups for nodal lines with
higher-order dispersions that can be stabilized by crystalline symmetry in
solid state systems with spin-orbit coupling and time reversal symmetry. We
find that besides conventional linear nodal lines, only lines with quadratic or
cubic dispersions are possible, for which the allowed degeneracy cannot be
larger than two. We derive effective Hamiltonians to characterize the novel
low-energy fermionic excitations for the quadratic and cubic nodal lines, and
explicitly construct minimal lattice models to further demonstrate their
existence. Their signatures can manifest in a variety of physical properties
such as the (joint) density of states, magneto-response, transport behavior,
and topological surface states. Using ab-initio calculations, we also identify
possible material candidates that realize these exotic nodal lines.
