  {\em Honeycomb toroidal graphs} are a family of cubic graphs determined by a
set of three parameters, that have been studied over the last three decades
both by mathematicians and computer scientists. They can all be embedded on a
torus and coincide with the cubic Cayley graphs of generalized dihedral groups
with respect to a set of three reflections. In a recent survey paper B. Alspach
gathered most known results on this intriguing family of graphs and suggested a
number of research problems regarding them. In this paper we solve two of these
problems by determining the full automorphism group of each honeycomb toroidal
graph.
