In this paper, we consider a scenario where an energy harvesting sensor continuously monitors a system and sends time-stamped status updates to a destination. The destination keeps track of the system status through the received updates. We use the metric Age of Information (AoI), the time that has elapsed since the last received update was generated, to measure the "freshness" of the status information available at the destination. We assume energy arrives randomly at the sensor according to a Poisson process, and each status update consumes one unit of energy. Our objective is to design optimal online status update policies to minimize the long-term average AoI, subject to the energy causality constraint at the sensor. We consider three scenarios, i.e., the battery size is infinite, finite, and one unit only, respectively. For the infinite battery scenario, we adopt a best-effort uniform status update policy and show that it minimizes the long-term average AoI. For the finite battery scenario, we adopt an energy-aware adaptive status update policy, and prove that it is asymptotically optimal when the battery size goes to infinity. For the last scenario where the battery size is one, we first show that within a broadly defined class of online policies, the optimal policy should have a renewal structure, i.e., the status update epochs form a renewal process, and the length of each renewal interval depends on the first energy arrival over that interval only. We then focus on a renewal interval, and prove that if the AoI in the system is below a threshold when the first energy arrives, the sensor should store the energy and hold status update until the AoI reaches the threshold, otherwise, it updates the status immediately. We analytically characterize the long-term average AoI under such a threshold-based policy, and explicitly identify the optimal threshold.