This paper addresses scheduling and memory management in input queued switches with finite input buffers, with the objective of minimizing packet loss. The framework and algorithms proposed here apply to buffer constrained wireless networks as well. The scheduling problem has been extensively addressed under the assumption of infinite input buffers. We study the finite buffer case here which arises in practice. The introduction of memory constraint significantly complicates the problem. The optimal strategies for infinite buffer case no longer apply and become strictly suboptimal in presence of memory limitations. We present closed form optimal strategies which minimize packet loss in 2 x 2 switches with equal arrival rates for all streams. We identify certain characteristics of the optimal strategy for arbitrary arrival rates, and use these properties to design a near optimal heuristic. We use the insight obtained from the investigation for 2 x 2 switches to propose a heuristic for N x N switches, arbitrary N and show numerically that this strategy performs close to optimal. The policies presented here reduce packet loss by about 25% as compared to the optimal strategy for the infinite buffer case.