Vector transmission-line theory is introduced for problems involving plane wave propagation, reflection and transmission in multilayered general bi-isotropic (BI) structures. The structure is replaced by a transmission line with vector voltage and current quantities and dyadic line parameters. The characteristic admittance and propagation factor are both dyadic quantities, which have different expressions for waves propagating in opposite directions along the transmission line. Vector circuit theory, developed earlier for plane-wave propagation and reflection problems in layered chiral and general BI structures, is formulated in terms of vector transmission line parameters. The theory is verified by checking special cases. It is claimed that, with the present formalism, analysis can be done more systematically, with less effort and particular results can be memorized more easily than when operating with scalar component quantities. Numerical examples show new and interesting results indicating possible applications of BI slabs.