We describe a new type of scanning electron microscope which works by directly imaging the electron field-emission sites on a nanotip. Electrons are extracted from the nanotip through a nanoscale aperture, accelerated in a high electric field and focussed to a spot using a microscale einzel lens. If the whole microscope (accelerating section and lens) and the focal length are both restricted in size to below 10 microns, then computer simulations show that the effects of aberration are extremely small and it is possible to have a system with approximately unit magnification, at electron energies as low as 300 eV. Thus a typical emission site of 1 nm diameter will produce an image of the same size and an atomic emission site with give a resolution of 0.1-0.2 nm (1-2 Å), and because the beam is not allowed to expand beyond 100nm in diameter the depth of field is large and the contribution to the beam spot size from chromatic aberrations is less than 0.02 nm (0.2 Å) for 500 eV electrons. Since it is now entirely possible to make stable atomic sized emitters (nanopyramids) it is expected that this instrument will have atomic resolution. Furthermore the brightness of the beam is determined only by the field-emission and can be up to a million times larger than in a typical (high-energy) electron microscope. The construction of this microscope, based on using a nanotip electron source which is mounted on a nanopositioner so that it can be positioned at the correct point adjacent to the microscope, entrance aperture, is described. In this geometry the scanning is achieved by moving the sample using piezos. Two methods for the construction of the microscope column are reviewed and the results of preliminary tests are described. The advantages of this low energy, bright-beam, electron microscope with atomic resolution are described. It can be used in either scanning mode or diffraction mode. The major advantage over existing microscopes is that because it works at very low energies the elastic backscattering is sensitive to the atomic species and so these can be identified directly without any energy discrimination on the detector. Furthermore it is also possible to use the microscope to do low energy electron diffraction which, because the scattering cross-section is large, can be carried out on single molecules. If these are biological samples such as DNA, proteins and viruses then the low energy means that the radiation damage is minimised. Some possibilities for mounting these samples, which can reduce radiation damage, are discussed. Finally we show a system for producing holograms of single protein molecules.