Abstract. Since Manski's (1975) seminal work, the maximum score method for discrete choice models has been applied to various econometric problems. Kim and Pollard (1990) established the cube root asymptotics for the maximum score estimator. Since then, however, econometricians posed several open questions and conjectures in the course of generalizing the maximum score approach, such as (a) asymptotic distribution of the conditional maximum score estimator for a panel data dynamic discrete choice model (Honoré and Kyriazidou, 2000), (b) convergence rate of the modified maximum score estimator for an identified set of parameters of a binary choice model with an interval regressor (Manski and Tamer, 2002), and (c) asymptotic distribution of the conventional maximum score estimator under dependent observations. To address these questions, this article extends the cube root asymptotics into four directions to allow (i) criterions drifting with the sample size typically due to a bandwidth sequence, (ii) partially identified parameters of interest, (iii) weakly dependent observations, and/or (iv) nuisance parameters with possibly increasing dimension. For dependent empirical processes that characterize criterions inducing cube root phenomena, maximal inequalities are established to derive the convergence rates and limit laws of the M-estimators. This limit theory is applied not only to address the open questions listed above but also to develop a new econometric method, the random coefficient maximum score. Furthermore, our limit theory is applied to address other open questions in econometrics and statistics, such as (d) convergence rate of the minimum volume predictive region (Polonik and Yao, 2000), (e) asymptotic distribution of the least median of squares estimator under dependent observations, (f) asymptotic distribution of the nonparametric monotone density estimator under dependent observations, and (g) asymptotic distribution of the mode regression and related estimators containing bandwidths.