We propose a new technique for the detection of single atoms in ultracold quantum gases. The technique is based on scanning electron microscopy and employs the electron impact ionization of trapped atoms with a focused electron probe. Subsequent detection of the resulting ions allows for the reconstruction of the atom's position. This technique is expected to achieve a much better spatial resolution compared to any optical detection method. In combination with the sensitivity to single atoms, it makes new in situ measurements of atomic correlations possible. The detection principle is also well suited for the addressing of individual sites in optical lattices.