Due to structural and aerodynamic limitations, conformal antennas are increasingly sought after in antenna design. Gap waveguide technology has also gained traction for its potential in leaky wave antenna development. This paper proposes a novel conformal leaky wave antenna utilizing this emerging gap waveguide technology. Prior research has not extensively explored the radiation characteristics of such antennas on curved and conformal surfaces. This work addresses this gap by investigating the design procedure, radiation mechanism, and optimization methods for leaky wave antennas on conformal surfaces. The proposed antenna achieves a gain of approximately 18 dB and a scanning angle ranging from 1 to 33 degrees within the frequency band of 9.5 GHz to 12 GHz. Fabrication and subsequent measurements of the antenna were conducted, with the results demonstrating good agreement with the simulations.