Modern biology increasingly relies on fluorescence microscopy, which is driving demand for smaller, lighter, and cheaper microscopes. However, traditional microscope architectures suffer from a fundamental tradeoff: as lenses become smaller, they must either collect less light or image a smaller field of view. To break this fundamental tradeoff between device size and performance, we present a new concept for 3D fluorescence imaging that replaces lenses with an optimized amplitude mask placed a few hundred microns above the sensor and an efficient algorithm that can convert a single frame of captured sensor data into high-resolution 3D images. The result is FlatScope: a lensless microscope that is scarcely larger than an image sensor (roughly 0.2 grams in weight and less than 1 mm thick) and yet able to produce micron-resolution, high-frame-rate, 3D fluorescence movies covering a total volume of several cubic millimeters. The ability of FlatScope to reconstruct full 3D images from a single frame of captured sensor data allows us to image 3D volumes roughly 40,000 times faster than a laser scanning confocal microscope. We envision that this new flat fluorescence microscopy paradigm will lead to implantable endoscopes that minimize tissue damage, arrays of imagers that cover large areas, and bendable, flexible microscopes that conform to complex topographies.