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As many sensor network applications require deployment in remote and hard-to-reach areas, it is critical to ensure that such networks are capable of operating unattended for long durations. Consequently, the concept of using nodes with energy replenishment capabilities has been gaining popularity. However, new techniques and protocols must be developed to maximize the performance of sensor networks with energy replenishment. Here, we analyze limits of the performance of sensor nodes with limited energy, being replenished at a variable rate. We provide a simple localized energy management scheme that achieves a performance close to that with an unlimited energy source and at the same time keeps the probability of complete battery discharge low. Based on the insights developed, we address the problem of energy management for energy-replenishing nodes with finite battery and finite data buffer capacities. To this end, we give an energy management scheme that achieves the optimal utility asymptotically while keeping both the battery discharge and data loss probabilities low.