The theory to analyze and design tapered conformal line-source leaky-wave antennas is described in this letter. The deformation in the radiation pattern associated to the bending of the antenna is accurately predicted, and useful expressions are derived to taper the leaky-mode complex propagation constant in order to restore the high-directive radiated beam scanning at the desired angle. The analysis method and the design approach are validated with full-wave simulations of a curved tapered leaky-wave antenna in hybrid technology.