A manifestly Lorentz invariant action is found for the Floreanini-Jackiw chiral boson. The method involves a novel chiral reduction of the phase-space action for a string and can be adapted to describe chiral bosons on the heterotic string worldsheet. A similar manifestly Lorentz invariant action is found for an entire class of conformal chiral 2k-form electrodynamics in (4k+2) dimensions which includes the Floreanini-Jackiw theory as the k=0 case.