  The natural automorphism group of a translation surface is its group of
translations. For finite translation surfaces of genus g > 1 the order of this
group is naturally bounded in terms of g due to a Riemann-Hurwitz formula
argument. In analogy with classical Hurwitz surfaces, we call surfaces which
achieve the maximal bound Hurwitz translation surfaces. We study for which g
there exist Hurwitz translation surfaces of genus g.
