The paper discusses the proposed Black Hole Explorer (BHEX) mission, which aims to detect and measure the shape of the "photon ring" around supermassive black holes M87* and Sgr A*. The photon ring is a thin, bright ring observed in black hole images, resulting from photons orbiting the black hole multiple times before escaping. This ring provides a direct probe of the Kerr geometry and its parameters, such as mass and spin. BHEX will use radio-interferometric observations at frequencies of 100 GHz to 300 GHz from an orbital distance of approximately 30,000 km. The mission will target the photon rings around M87* and Sgr A*, enabling measurements that confirm the Kerr nature of these sources and provide precise estimates of their masses and spins. The paper reviews the theoretical background of the photon ring, including its structure in the image domain and its interferometric signature in the visibility domain. It also discusses the expected precision of spin measurements for Sgr A* and the challenges and prospects for photon ring science with BHEX.The paper discusses the proposed Black Hole Explorer (BHEX) mission, which aims to detect and measure the shape of the "photon ring" around supermassive black holes M87* and Sgr A*. The photon ring is a thin, bright ring observed in black hole images, resulting from photons orbiting the black hole multiple times before escaping. This ring provides a direct probe of the Kerr geometry and its parameters, such as mass and spin. BHEX will use radio-interferometric observations at frequencies of 100 GHz to 300 GHz from an orbital distance of approximately 30,000 km. The mission will target the photon rings around M87* and Sgr A*, enabling measurements that confirm the Kerr nature of these sources and provide precise estimates of their masses and spins. The paper reviews the theoretical background of the photon ring, including its structure in the image domain and its interferometric signature in the visibility domain. It also discusses the expected precision of spin measurements for Sgr A* and the challenges and prospects for photon ring science with BHEX.