Age of information (AoI) is a recently introduced metric that measures the freshness of status updates in monitoring applications and update-based systems. In this thesis, we investigate a wireless communication system in which the status updates of the underlying process of interest can be sampled at any time by the source node, and are transmitted to a destination node over an error-prone communication channel under transmission constraints. More specifically, we study the scheduling of sampling and transmission of status updates in order to minimize the long-term average AoI at the destination for under a constraint on the average number of transmissions, or under energy replenishment constraints at the source. After each transmission, the source receives an instantaneous ACK/NACK feedback, and judiciously decides on the next update, without prior knowledge on the success of future transmissions. The optimal scheduling policy and its structure are studied under different feedback mechanisms when the system characteristics are known; in particular, the standard automatic repeat request (ARQ) and hybrid ARQ (HARQ) protocols are considered. In addition, average-cost reinforcement learning (RL) algorithms are proposed for practical applications when the system characteristics are not known in advance and must be learned in real time. Extensive simulation results are provided to demonstrate the effect of feedback, HARQ protocol, and resource constraints on the average AoI.