It is now widely recognized that increased levels of parallelism is a necessary condition for improved application performance on multicore computers. However, as the number of cores increases, the memory-per-core ratio is expected to further decrease, making per-core memory efficiency of parallel programs an even more important concern in future systems. For many parallel applications, the memory requirements can be significantly larger than for their sequential counterparts and, more importantly, their memory utilization depends critically on the schedule used when running them. To address this problem we propose bounded memory scheduling (BMS) for parallel programs expressed as dynamic task graphs, in which an upper bound is imposed on the program’s peak memory. Using the inspector/executor model, BMS tailors the set of allowable schedules to either guarantee that the program can be executed within the given memory bound, or throw an error during the inspector phase without running the computation if no feasible schedule can be found. Since solving BMS is NP-hard, we propose an approach in which we first use our heuristic algorithm, and if it fails we fall back on a more expensive optimal approach which is sped up by the best-effort result of the heuristic. Through evaluation on seven benchmarks, we show that BMS gracefully spans the spectrum between fully parallel and serial execution with decreasing memory bounds. Comparison with OpenMP shows that BMS-CnC can execute in 53% of the memory required by OpenMP while running at 90% (or more) of OpenMP’s performance.