  The spin point groups are finite groups whose elements act on both real space
and spin space. Among these groups are the magnetic point groups in the case
where the real and spin space operations are locked to one another. The
magnetic point groups are central to magnetic crystallography for strong
spin-orbit coupled systems and the spin point groups generalize these to the
intermediate and weak spin-orbit coupled cases. The spin point groups were
introduced in the 1960's and enumerated shortly thereafter. In this paper, we
complete the theory of spin point groups by presenting an account of these
groups and their representation theory. Our main findings are that the
so-called nontrivial spin point groups (numbering 598 groups) have co-irreps
corresponding exactly to the (co-)irreps of regular or black and white groups
and we tabulate this correspondence for each nontrivial group. However a total
spin group, comprising the product of a nontrivial group and a spin-only group,
has new co-irreps in cases where there is continuous rotational freedom. We
provide explicit co-irrep tables for all these instances. We also discuss new
forms of spin-only group extending the Litvin-Opechowski classes. To exhibit
the usefulness of these groups to physically relevant problems we discuss a
number of examples from electronic band structures of altermagnets to magnons.
