Empirical economic research frequently applies maximum likelihood estimation in cases where the likelihood function is analytically intractable. Most of the theoretical literature focuses on maximum simulated likelihood (MSL) estimators, while empirical and simulation analyzes often find that alternative approximation methods such as quasi-Monte Carlo simulation, Gaussian quadrature, and integration on sparse grids behave considerably better numerically. This paper generalizes the theoretical results widely known for MSL estimators to a general set of maximum approximated likelihood (MAL) estimators. We provide general conditions for both the model and the approximation approach to ensure consistency and asymptotic normality. We also show specific examples and finite-sample simulation results.