In this paper we formulate the theory of tunneling into general Abelian fra
ctional quantum Hall edge states. In contrast to the simple Laughlin starts
, a number of charge transfer processes must be accounted for. Nonetheless,
it is possible to identify a unique value corresponding to dissipationless
transport as the asymptotic large-V conductance through a tunneling juncti
on, and find fixed points (conformal held theory boundary conditions) corre
sponding to this value. The symmetries of a given edge tunneling problem de
termine the appropriate boundary condition, and the boundary condition dete
rmines the strong-coupling operator content and current noise.