A design of high-gain Fabry-Perot cavity (FPC) antenna is presented with non-uniform frequency selective surface (FSS) superstrate and electromagnetic band gap (EBG) reflecting ground. The non-uniform FSS superstrate and EBG ground are utilized to flexibly control the attenuation constant $\alpha $ over a large antenna aperture while ensuring the resonance condition of the FPC. The uniformity of the amplitude of the aperture field is improved significantly while obtaining appropriate radiation efficiency. Therefore, the high gain and high aperture efficiency can be achieved within a desirable bandwidth. The theoretical gain estimation is conducted to prove the superiority of the proposed antenna over a referenced uniform one. A straight-forward design methodology is proposed based on the radial leaky-wave theory and the modified transverse equivalent network (TEN) model. The non-uniform high-gain FPC antenna with a diameter of $6.6\lambda _{0}$ is designed at the frequency of 15 GHz. Simulated results show that a gain of 24.6 dBi with an aperture efficiency of 67.9% has been achieved. The measured gain is 24.0 dBi with an aperture efficiency of 58.4%. The 10dB return loss bandwidth and 3 dB gain bandwidth are 5.3% and 3.2%, respectively. The proposed antenna overcomes the limit on aperture efficiency and gain of conventional uniform FPC antennas, and the presented methodology can guide the design of two-dimensional non-uniform high-gain FPC antennas.