As Ernst Abbe showed in 1870, the resolution of a light microscope is given by the ratio of wavelength to numerical aperture. This limit is about 200 nm, which can only be further reduced by decreasing the wavelength. That's why electrons enter the play.
Because of the dualism between particles and waves, which was discovered by Louis de Broglie in 1924, each electron can be assigned a wavelength. This wavelength depends only on the electron's energy. Since the electron can be accelerated by application of a voltage, its wavelength can be decreased far beyond the shortest visible wavelength of light. Theoretically a resolution of 0.037 Å at an acceleration voltage of 100 kV can be achieved. Instead of glass lenses rotational-symmetric electromagnetic fields have to be used to focus the electron beam. But due to intrinsic aberrations of these lenses the obtainable resolution is about 100 times worse than the theoretical limit.
In order to approach this theoretical limit, correctors have to be introduced into the column. These are multipole elements which compensate for the chromatic and/or spherical aberrations of the lenses. The resolution limit on a 200 kV TEM is about 2 Å without corrector and 0.6 Å with corrector.
Further pushing the resolution limit by increasing the electron energy is possible, but the risk of damaging the investigated specimen has to be considered.
Electron microscopes are nowadays indispensable instruments in scientific research. Here are some examples for their usefulness: