A new form of coherent optical microscopy is described that generates images (reconstructions) of two and three-dimensional, penetrable scattering objects computationally from sets of measured digital holograms of scattered field data collected in a suite of scattering experiments. The microscope uses the technique of "phase shifting holography" (explained in the talk) to compute both the amplitude and phase of the coherently scattered field data which is then input into wave based inversion algorithms to generate the image of the object. For 2D (thin) objects the inversion algorithm is based on a coherent back propagation of field data collected in single experiment and, unlike conventional microscopy, yields both the amplitude as well as phase of the 2D sample. For thick (3D) objects the object is embedded in an index matching fluid and a suite of scattering experiments is required using dirent incident illuminating waves. In this case the object reconstruction is generated using a "generalized" filtered back propagation (FBP) algorithm that allows for multiple scattering between the object and index matching bath. The generalized FBP algorithm is described and examples using simulated and real data are presented.