The SWEEPS (Sagittarius Window Eclipsing Extrasolar Planet Search) program was aimed at detecting planets around stars in the Galactic bulge, not only to determine their physical properties, but also to determine whether the properties of planets found in the solar neighborhood, such as their frequency and the metallicity dependence, also hold for the planets in the Galactic bulge. We used the Hubble Space Telescope to monitor 180,000 F, G, K, and M dwarfs in the Galactic bulge continuously for 7 days in order to look for transiting planets. We discovered 16 candidate transiting extrasolar planets with periods of 0.6 to 4.2 days, including a possible new class of ultra-short period planets (USPPs) with P < 1 day. The facts that (i) the coverage in the monitoring program is continuous, (ii) most of the stars are at a known distance (in the Galctic bulge), (iii) monitoring was carried out in 2 passbands, and (iv) the images have high spatial resolution, were crucial in minimizing and estimating the false positive rates. We estimate that at least 45% of the candidates are genuine planets. Radial velocity observations of the two brightest host stars further support the planetary nature of the transiting companions. These results suggest that the planet frequency in the Galactic bulge is similar to that in the solar neighborhood. They also suggest that higher metallicity favors planet formation even in the Galactic bulge. The USPPs occur only around low-mass stars which may suggest that close-in planets around higher-mass stars are irradiately evaporated, or that planets are able to migrate to and survive in close-in orbits only around such old and low-mass stars.