  Spin waves in antiferromagnetic materials have great potential for
next-generation magnonic technologies. However, their properties and their
dependence on the type of ground-state antiferromagnetic structure are still
open questions. Here, we investigate theoretically spin waves in one- and
two-dimensional model systems with a focus on noncollinear antiferromagnetic
textures such as spin spirals and skyrmions of opposite topological charges. We
address in particular the nonreciprocal spin excitations recently measured in
bulk antiferromagnet $\alpha$--$\text{Cu}_2\text{V}_2\text{O}_7$ utilizing
inelastic neutron scattering experiments [Phys.\ Rev.\ Lett.\ \textbf{119},
047201 (2017)], where we help to characterize the nature of the detected
spin-wave modes. Furthermore, we discuss how the Dzyaloshinskii-Moriya
interaction can lift the degeneracy of the spin-wave modes in antiferromagnets,
resembling the electronic Rashba splitting. We consider the spin-wave
excitations in antiferromagnetic spin-spiral and skyrmion systems and discuss
the features of their inelastic scattering spectra. We demonstrate that
antiskyrmions can be obtained with an isotropic Dzyaloshinskii-Moriya
interaction in certain antiferromagnets.
