Based on an original numerical reconstruction algorithm (E. Cuche et al. Appl. Opt. 38, 6994 1999), we have developed a Digital Holographic Microscope (DHM), in a transmission mode, allowing to investigate noninvasively cellular structures and dynamics. DHM images of living cells in culture are presented. They represent the distribution of the optical path length over the cell, which contains information concerning both the cellular morphometry and the intracellular refractive index, and which has been measured with a sub-wavelength accuracy.