The characteristic impedance of a microwave transmission line is typically constrained to a value , in part because of the low impedance of free space and the limited range of permittivity and permeability realizable with conventional materials. Here we suggest the possibility of constructing high-impedance transmission lines by exploiting the plasmonic response of edge states associated with the quantum Hall effect in gated devices. We analyze various implementations of quantum Hall transmission lines based on distributed networks and lumped-element circuits, including a detailed account of parasitic capacitance and Coulomb drag effects, which can modify device performance. We additionally conceive of a metamaterial structure comprising arrays of quantum Hall droplets and analyze its unusual properties. The realization of such structures holds promise for efficiently wiring-up quantum circuits on chip, as well as engineering strong coupling between semiconductor qubits and microwave photons.