We describe a holographic microscope with a spatial resolution approaching the diffraction limit. The instrument uses a tiny drop of glycerol as a lens to create the spherically diverging reference illumination necessary for Fourier-transform holography. Measurement of the point-spread function, which is obtained by imaging a knife edge in dark-field illumination, indicates a transverse resolution of 1.4 microm with wavelength lambda = 514.5 nm. Longitudinal resolution is obtained from the holograms by the numerical equivalent of optical sectioning. We describe the method of reconstruction and demonstrate the microscope's capability with selected biological specimens. The instrument offers two unique capabilities: (1) it can collect three-dimensional information in a single pulse of light, avoiding specimen damage and bleaching; and (2) it can record three-dimensional motion pictures from a series of light pulses. The conceptual design is applicable to a broad range of wavelengths and we discuss extension to the x-ray regime.