The methodology to approach the quantum-classical correspondence problem is reviewed. Although decoherence seems to be the most promising way to understand the quantum-classical correspondence, one in the delta kicked rotator(DKR) is not fully explained by decoherence. It is shown that the unrealistic nature of the delta kick is responsible for this failure. The Gaussian kicked rotator(GKR) is introduced as a realistic version of DKR, and the quantum-classical correspondence in GKR is discussed at the levels of both expectation values and phase space distributions.