We present in this talk some profound consequences resulting from non-locality and non-linearity of Hartree-Fock (HF) equations that describe most accurately many-body systems in one-particle approximation. It is of special interest since these equations penetrate all physics, from nuclei to condensed matter. In 1930 V. A. Fock has corrected non-linear D. Hartree’s equations (1928) by introducing an exchange term that eliminated self-action and restored orthogonality of states with different energies. This term led to non-locality of the effective one-particle potential and preserve non-linearity. It appeared that non-locality has a number of essential consequences. Here these consequences are listed:

1. Solutions of HF equations have extra zeroes, even for the lowest 1s level, if states with higher principal quantum numbers are occupied. The number of zeroes is not totally determined by the radial quantum number of the considered level, but mainly by the radial quantum number of the outermost particle.
2. The asymptotic of any one-electron HF occupied state wave function is determined not by the state’s binding energy but by the energy of the outermost particle. This profoundly increases the probability of ionization of the inner levels in strong laser fields.
3. The HF equations are gauge-non-invariant, so the so-called “length” and “velocity” forms of the interaction between electromagnetic wave and the considered system’s particles are non-equivalent and the dipole sum rule is violated. To restore gauge-invariance one has to go beyond HF, namely to the Random Phase Approximation with Exchange, or to the Time-Dependent HF. Note that the width of the Giant resonance in photo-absorption cross-section of the system is determined mainly by non-locality.
4. The expression for the one-particle Green’s function via product of regular and irregular solutions of the HF equations is incorrect.
5. The scattering Levinson’s theorem is violated so that the phase shift at zero energy is determined by the number of not only vacant but also occupied bound states.
6. The exchange between incoming and atomic electrons combined with Coulomb nature of interelectron interaction leads to complex poles in the electron-atom scattering amplitude, located at the binding energy of the electron in the target atom, not the “electron + atom” system
7. Non-linearity leads to non-uniqueness of HF solutions for the same energy. This is demonstrated in the frame of two models for inter-particle interaction.
8. Points 3 and 4 are valid not only for HF but for any one-particle non-local potential.

HF equations were studied and applied mainly to multi-fermion systems. For bosons the exchange correction does not eliminate self-action, leading instead to doubling of its effect.