Many of the problems of transmission electron microscopy (TEM) are due to the fact that wave optics which governs the interaction of electrons with the specimen and the imaging process definitely is brought to an end with the detection of the final electron image. Unfortunately, resolution is limited by an increasing number of aberrations. Furthermore, wave optical tools in the electron microscope which are needed for example to produce phase contrast better than that given by the phase contrast transfer function, for distinction of amplitude contrast and phase contrast, or to measure phases in Fourier space, are only poorly developed. Since subsequent image processing of electron diffraction patterns or real space images can never compensate for the loss of phase information, the phase has to be recorded also, i.e. one has to work holographically to collect and reconstruct all data of the object structure.