The main purpose of this research is to evaluate the performance of a Bayesian approach for estimating unknown change points using Monte Carlo simulations. The univariate and bivariate unknown change point mixed models were presented and the basic idea of the Bayesian approach for estimating the models was discussed. The performance of Bayesian estimation was evaluated using simulation studies of longitudinal data with different sample sizes, varying change point values, different levels of Level-1 variances, and univariate versus bivariate outcomes. The numerical results compared the performance of the Bayesian methods with the first-order Taylor expansion method and the adaptive Gaussian quadrature method implemented in SAS PROC NLMIXED. These simulation results showed that the first-order Taylor expansion method and the adaptive Gaussian quadrature method were sensitive to the initial values, making the results somewhat unreliable. In contrast, these simulation results showed that Bayesian estimation was not sensitive to the initial values and the fixed-effects and Level-1 variance parameters can be accurately estimated in all of the conditions. One concern was that the estimates of the Level-2 covariance parameters were found to be biased when the Level-1 variance was large in the bivariate model. However, and in general, the new Bayesian approach to the estimation of turning points in longitudinal data proved to be quite robust and practically useful. (Contains 7 tables, 2 figures, and 2 notes.)