The complex Langevin method (CLM) is one of the promising approaches to overcome the sign problem in conventional Monte Carlo simulations and is expected to reveal the phase diagram of QCD in the high-density region. On the other hand, the CLM is reliable only when the probability distribution of the drift term falls off exponentially or faster. Whether the CLM is reliable or not in the high-density region is still an open problem. In this talk, I will present recent results on the physics of QCD at finite density, based on the reliability of the CLM. I will also discuss possible setups to realize color superconductivity for future CLM simulations.