Ranking and selection (R&S) procedures are widely used for selecting the best among a set of candidate systems, where each candidate system is associated with a simulation model. In this thesis, we focus on three aspects on the sample com- plexity of the R&S problem. First, we develop a method for predicting the sample complexity. Second, we present Envelope Procedure (EP), a R&S procedure that delivers a probably approximately correct selection guarantee, and we provide a high probability upper bound on its sample complexity. We also prove a lower bound on the sample complexity for general R&S procedures. The performance of the EP is demonstrated by numerical experiments. Finally, we discuss some specific aspects and features of the EP in parallel computing environment and the sampling rules.