In 1970, Breedlove and Trammel predicted that, because of radiation damage, the direct imaging of individual organic molecules by electron or X-ray microscopy was not possible. This follows from an analysis, above IkeV, of the beam energy dependence of the ratio of inelastic scattering cross sections (some of which causes damage) to elastic cross sections (used to form the image). To minimize damage, we have therefore constructed a transmission electron microscope with nanometer resolution, which operates at 7 - 300 eV, below the threshold for most inelastic processes. Recently we have obtained our first images of TMV with this instrument. Isaacson first suggested that the K-shell ionization of carbon may be the most important damage process. Although less probable than valence excitation, it dumps more energy into the system, and the decay products are highly damaging. A subsequent study of electron beam damage in several π- bonded hydrocarbon crystals down to about 200 eV showed that a damage threshold does exist at about IkeV incident energy which can be explained by the carbon K-shell ionization. (Multiple scattering increases the energy threshold from 285eV).