  We study 5-dimensional Riemannian manifolds that admit an almost contact
metric structure. We classify these structures by their intrinsic torsion and
review the literature in terms of this scheme. Moreover, we determine necessary
and sufficient conditions for the existence of metric connections with
vectorial, totally skew-symmetric or traceless cyclic torsion that are
compatible with the almost contact metric structure. Finally, we examine
explicit examples of almost contact metric 5-manifolds from this perspective.
