We report on new developments of a technique, called Digital Holographic Microscopy (DHM), for the numerical reconstruction of digital holograms taken in microscopy, which allows simultaneous amplitude and quantitative phase contrast imaging. The reconstruction method computes the propagation of the complex optical wavefront diffracted by the object and is used to determine the refractive index and/or shape of the object with accuracies in the nanometer range along the optical axis. The method requires the acquisition of a single hologram. The technique comprises the recording of a digital hologram of the specimen by means of a standard CCD camera at the exit of a Mach-Zender or Michelson type interferometer. The quantitative nature of the reconstructed phase distribution has been demonstrated by an application to surface profilometry where step height differences of a few nanometers have been measured. Another application takes place in biology for transmission phase-contrast imaging of living cells in culture. The resolution for thickness measurements depends on the refractive index of the specimen and a resolution of approximately 30 nanometers in height, and about half of a micron in width, has been achieved for living neural cells in cultures by using a high numerical aperture.