  Band topology is both constrained and enriched by the presence of symmetry.
The importance of anti-unitary symmetries such as time reversal was recognized
early on leading to the classification of topological band structures based on
the ten-fold way. Since then, lattice point group and non-symmorphic symmetries
have been seen to lead to a vast range of possible topologically nontrivial
band structures many of which are realized in materials. In this paper we show
that band topology is further enriched in many physically realizable instances
where magnetic and lattice degrees of freedom are wholly or partially
decoupled. The appropriate symmetry groups to describe general magnetic systems
are the spin-space groups. Here we describe cases where spin-space groups are
essential to understand the band topology in magnetic materials. We then focus
on magnon band topology where the theory of spin-space groups has its simplest
realization. We consider magnetic Hamiltonians with various types of coupling
including Heisenberg and Kitaev couplings revealing a hierarchy of enhanced
magnetic symmetry groups depending on the nature of the lattice and the
couplings. We describe, in detail, the associated representation theory and
compatibility relations thus characterizing symmetry-enforced constraints on
the magnon bands revealing a proliferation of nodal points, lines, planes and
volumes.
