  We apply the geometric theory of swimming at low Reynolds number to the study
of nearly circular swimmers in two-dimensional fluids with non-vanishing Hall,
or "odd", viscosity. The Hall viscosity gives an off-diagonal contribution to
the fluid stress-tensor, which results in a number of striking effects. In
particular, we find that a swimmer whose area is changing will experience a
torque proportional to the rate of change of the area, with the constant of
proportionality given by the coefficient $\eta^o$ of odd viscosity. After
working out the general theory of swimming in fluids with Hall viscosity for a
class of simple swimmers, we give a number of example swimming strokes which
clearly demonstrate the differences between swimming in a fluid with
conventional viscosity and a fluid which also has a Hall viscosity. A number of
more technical results, including a proof of the torque-area relation for
swimmers of more general shape, are explained in a set of appendices.
