In this paper we provide faster algorithms for approximately solving discounted Markov Decision Processes in multiple parameter regimes. Given a discounted Markov Decision Process (DMDP) with |S| states, |A| actions, discount factor γ ϵ (0; 1), and rewards in the range [-M;M], we show how to compute an ϵ-optimal policy, with probability 1 -δ in time1 Õ(( |S|2|A| + |S||A|/(1 -γ )3) log (M/ϵ) log (1/δ)) : This contribution reects the first nearly linear time, nearly linearly convergent algorithm for solving DMDP's for intermediate values of. We also show how to obtain improved sublinear time algorithms and provide an algorithm which computes an ϵ-optimal policy with probability 1-δ in time Õ (|S||A|M2/(1-γ)4ϵ2 log ( 1/δ)) provided we can sample from the transition function in O(1) time. Interestingly, we obtain our results by a careful modification of approximate value iteration. We show how to combine classic approximate value iteration analysis with new techniques in variance reduction. Our fastest algorithms leverage further insights to ensure that our algorithms make monotonic progress towards the optimal value. This paper is one of few instances in using sampling to obtain a linearly convergent linear programming algorithm and we hope that the analysis may be useful more broadly.