Abstract: This talk is a survey of recent work in power saving strategies. We address two different mechanisms for saving power. The first is the ability to transition a device or system into a sleep state if it is idle. The trade-off is that a fixed amount of energy is needed to transition the device back into an active state in which it can begin to process jobs. We extend previous work on this problem and develop online algorithms for devices with multiple levels of sleep states. Each sleep state has a different power consumption level and different cost in transitioning back to the active state. The second way in which power savings can be achieved is by varying the speed at which jobs are run. We utilize a power consumption curve which indicates the power consumption level given a particular speed. The problem is to schedule arriving jobs in a way that minimizes total energy use so that each job is completed by its deadline. Our work addresses systems whose speed can be varied and which have a sleep state. We devise both offline and online algorithms for this problem.