While the asymptotic normality of the maximum likelihood estimator under regularity conditions is long established, this paper derives explicit bounds for the bounded Wasserstein distance between the distribution of the maximum likelihood estimator (MLE) and the normal distribution. For this task, we employ Stein's method. We focus on independent and identically distributed random variables, covering both discrete and continuous distributions as well as exponential and non-exponential families. In particular, a closed form expression of the MLE is not required. We also use a perturbation method to treat cases where the MLE has positive probability of being on the boundary of the parameter space.