I know that on an algebraic function field in one variable over any base field, there is a good divisor theory for it and a Riemann-Roch Theorem; in particular, there is a 'good' notion of 'genus'. (Which, I guess, is in general different from the corresponding notions in the geometric sense.) References are e.g. Chevalley's book and Deuring's book on algebraic function field in one variable, and Fried&Jarden's "Field Arithmetic".

Is there a generalization of these things to an arbitrary function field, which is of arbitrary transcendental degree over an arbitrary base field? Thanks!