  We discuss in a systematic way the gauge theory for a continuous spin
particle proposed by Schuster and Toro. We show that it is naturally formulated
in a cotangent bundle over Minkowski spacetime where the gauge field depends on
the spacetime coordinate ${x^\mu}$ and on a covector $\eta_\mu$. We discuss how
fields can be expanded in $\eta_\mu$ in different ways and how these expansions
are related to each other. The field equation has a derivative of a Dirac delta
function with support on the $\eta$-hyperboloid $\eta^2+1=0$ and we show how it
restricts the dynamics of the gauge field to the $\eta$-hyperboloid. We then
show that on-shell the field carries one single irreducible unitary
representation of the Poincar\'e group for a continuous spin particle. We also
show how the field can be used to build a set of covariant equations found by
Wigner describing the wave function of one-particle states for a continuous
spin particle. Finally we show that it is not possible to couple minimally a
continuous spin particle to a background abelian gauge field, and make some
comments about the coupling to gravity.
