The exoplanets discovered so far have been mostly around relatively nearby and bright stars. As a result, the host stars are mostly (i) in the Galactic disk, (ii) relatively massive, and (iii) relatively metal rich. The aim of the SWEEPS project is to extend our knowledge to stars which (i) are in a different part of the Galaxy, (ii) have lower masses, and (iii) have a large range of metallicities. To achieve this goal, we used the Hubble Space Telescope to search for transiting planets around F, G, K, and M dwarfs in the Galactic bulge. We photometrically monitored 180,000 stars in a dense bulge field continuously for 7 days. We discovered 16 candidate transiting extrasolar planets with periods of 0.6 to 4.2 days, including a new class of ultra-short period planets (USPPs) with P < 1.2 days. Radial-velocity observations of the two brightest candidates support their planetary nature. These results suggest that planets are as abundant in the Galactic bulge as they are in the solar neighborhood, and they are equally abundant around low-mass stars (within a factor 2). The planet frequency increases with metallicity even for the stars in the Galactic bulge. All the USPP hosts are low-mass stars, suggesting either that close-in planets around higher-mass stars are irradiatively evaporated, or that the planets can migrate to close-in orbits only around such old and low-mass stars.