The theory of electron momentum spectroscopy (EMS), a technique based on th
e single (multiple) electron impact ionization of a quantum target (atom, m
olecule, thin Nm), is reviewed. The advantages of EMS as a means for interp
reting outgoing momentum distributions as compared to other methods for dir
ectly measuring the many-body wave function of the target are vastly illust
rated. It is shown that EMS contributes profoundly to advances in the quant
um chemistry of molecules allowing, in particular, to gain insight into how
electron orbitals transform as molecules approach one another adiabaticall
y, and to study in detail the band structure and one-electron wave function
s in both crystalline and amorphous thin films. The possibility of the dire
ct double ionization "portraying" of the electron correlations is also stud
ied.