  In the present review we discuss different aspects of the two-photon exchange
(TPE) physics in elastic $ep$ scattering, at high $Q^2$ as well as at low
$Q^2$. The imaginary part of the TPE amplitude gives rise to beam and target
single-spin asymmetries. Different theoretical approaches to calculation of
these observables are considered. The real part of the TPE amplitude influence
unpolarized cross section and double-spin observables and is, most likely,
responsible for discrepancy between two methods of proton form factors
measurements. We review different methods of calculations of the TPE amplitudes
the framework of "hadron" and "quark-gluon" approaches. We discuss the
dispersion approach suitable for low and intermediate $Q^2$, which includes
elastic and inelastic intermediate hadronic states, as well as connection of
TPE to proton radius puzzle. The present situation with direct experimental
searches for the TPE amplitude in $e^+p/e^-p$ charge asymmetry is also
discussed, as well as attempts to extract the TPE amplitudes from existing
experimental data obtained by the Rosenbluth and double polarization
techniques. The TPE physics in other processes, such as elastic $\mu p$,
$e$-nucleus and $e\pi$ scattering is also reviewed.
