The acceleration of electrons by laser beams in vacuum is studied. Ele
ctrons can gain energy out of the laser field if they are accelerated
inside only a half wavelength of the laser light. With presently avail
able lasers one can reach final electron energies of up to about 100 M
eV. Conditions for TeV electron accelerators are elaborated. For study
ing the basic mechanism we suggest an experiment by crossing an electr
on beam with a laser beam.