  This article describes cubic function fields $L/K$ with prescribed
ramification, where $K$ is a rational function field. We give general equations
for such extensions, an explicit procedure to obtain a defining equation when
the purely cubic closure $K'/K$ of $L/K$ is of genus zero, and a description of
the twists of $L/K$ up to isomorphism over $K$. For cubic function fields of
genus at most one, we also describe the twists and isomorphism classes obtained
when one allows M\"obius transformations on $K$. The article concludes by
studying the more general case of covers of elliptic and hyperelliptic curves
that are ramified above exactly one point.
