  Open problems in the interpretation of the observed pair of near degenerate
$\Delta I = 1$ bands with the same parity as the chiral doublet bands are
discussed. The ambiguities for the existing fingerprints of the chirality in
atomic nuclei and problems in existing theory are discussed, including the
description of quantum tunneling in the mean field approximation as well as the
deformation, core polarization and configuration of particle rotor model (PRM).
Future developments of the theoretical approach are prospected.
