This paper studies the limit distributions of Monte Carlo estimators of diffusion processes. Two types of estimators are examined. The first one is based on the Euler scheme applied to the original processes; the second applies the Euler scheme to a variance-stabilizing transformation of the processes. We show that the transformation increases the speed of convergence of the Euler scheme. The limit distribution of this estimator is derived in explicit form and is found to be non-centered. We also study estimators of conditional expectations of diffusions with known initial state. Expected approximation errors are characterized and used to construct second-order bias corrected estimators. Such bias correction eliminates the size distortion of asymptotic confidence intervals and allows to examine the relative efficiency of estimators. Finally, we derive the limit distributions of Monte Carlo estimators of conditional expectations with unknown initial state. The variance-stabilizing transformation is again found to increase the speed of convergence. For comparison we also study the Milshtein scheme. We derive new convergence results for this scheme and show that it does not improve on the convergence properties of the Euler scheme with transformation. Our results are illustrated in the context of a dynamic portfolio choice problem and of simulated-based estimation of diffusion processes.