Extending the Monte Carlo method to dynamic critical phenomena we investigated the time-dependent correlation functions in the two-dimensional one-spin-flip Ising model and the critical behavior of the associated relaxation times. These relaxation times are the following: τδμΔT, characterizing the approach of the order parameter to equilibrium after a change of temperature ΔT of the system; τδμδμ and τδμδμA characterizing the slowing down of the order-parameter correlation and autocorrelation functions, respectively; τδHδH and τδHδHA, characterizing the slowing down of the energy correlation and autocorrelation functions; and finally τδμδH, characterizing the cross-correlation function. We give estimates for the associated exponents ΔδμΔTΔδμδμΔδHδ HΔδμδH1.90±0.10, and ΔδμδμA1.60±0.10, ΔδμδHA0.95±0.10, ΔδHδHA0, which are consistent with the dynamic scaling hypothesis and with exact inequalities. A detailed comparison with recent high-temperature-expansion estimates is performed, and the reliability of the Monte Carlo results is carefully analyzed. © 1973 The American Physical Society.