The image blurring by the Earth's atmosphere generally poses a substantial limitation to ground-based observations. While opportunities in space are scarce, lucky imaging can correct over a much larger patch of sky and with much fainter reference stars. We propose the first of a new kind of versatile instruments, "GravityCam", composed of ~100 EMCCDs, that will open up two entirely new windows to ground-based astronomy: (1) wide-field high-resolution imaging, and (2) wide-field high-speed photometry. Potential applications include (a) a gravitational microlensing survey going 4 magnitudes deeper than current efforts, and thereby gaining a factor 100 in mass at the same sensitivity, which means probing down to Lunar mass or even below, (b) extra-solar planet hunting via transits in galactic bulge fields, with high time resolution well-suited for transit timing variation studies, (c) variable stars in crowded fields, with sensitivity to very short periods, (d) asteroseismology with many bright stars in one pointing, (e) serendipitous occultations of stars by small solar system bodies, giving access to the small end of the Kuiper Belt size distribution and potentially leading to the first detection of true Oort cloud objects, while predicted occultations at high time resolution can reveal atmospheres, satellites, or rings, (f) general data mining of the high-speed variable sky (down to 40 ms cadence).