Two photon exchange in elastic electron scattering off hadronic systems

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.