High-resolution optical microscopy has traditionally relied on high-magnification and high-numerical aperture objective lenses. In contrast, lensless microscopy can provide high-resolution images without the use of any focusing lenses, offering the advantages of a large field of view, high resolution, cost-effectiveness, portability, and depth-resolved three-dimensional (3D) imaging. Here we review various approaches to lensless imaging, as well as its applications in biosensing, diagnostics, and cytometry. These approaches include shadow imaging, fluorescence, holography, superresolution 3D imaging, iterative phase recovery, and color imaging. These approaches share a reliance on computational techniques, which are typically necessary to reconstruct meaningful images from the raw data captured by digital image sensors. When these approaches are combined with physical innovations in sample preparation and fabrication, lensless imaging can be used to image and sense cells, viruses, nanoparticles, and biomolecules. We conclude by discussing several ways in which lensless imaging and sensing might develop in the near future.