We are developing a transmission electron microscope that operates at extremely low electron energies, 0-40. eV. We call this technique eV-TEM. Its feasibility is based on the fact that at very low electron energies the number of energy loss pathways decreases. Hence, the electron inelastic mean free path increases dramatically. eV-TEM will enable us to study elastic and inelastic interactions of electrons with thin samples. With the recent development of aberration correction in cathode lens instruments, a spatial resolution of a few nm appears within range, even for these very low electron energies. Such resolution will be highly relevant to study biological samples such as proteins and cell membranes. The low electron energies minimize adverse effects due to radiation damage.