  We consider the problem of which distance-regular graphs with small valency
are Cayley graphs. We determine the distance-regular Cayley graphs with valency
at most $4$, the Cayley graphs among the distance-regular graphs with known
putative intersection arrays for valency $5$, and the Cayley graphs among all
distance-regular graphs with girth $3$ and valency $6$ or $7$. We obtain that
the incidence graphs of Desarguesian affine planes minus a parallel class of
lines are Cayley graphs. We show that the incidence graphs of the known
generalized hexagons are not Cayley graphs, and neither are some other
distance-regular graphs that come from small generalized quadrangles or
hexagons. Among some ``exceptional'' distance-regular graphs with small
valency, we find that the Armanios-Wells graph and the Klein graph are Cayley
graphs.
