An agent facing sequential decisions that are characterized by partial feedback needs to strike a balance between maximizing immediate payoffs based on available information, and acquiring new information that may be essential for maximizing future payoffs. This trade-off is captured by the multi-armed bandit (MAB) framework that has been studied and applied when at each time epoch payoff observations are collected on the actions that are selected at that epoch. In this paper we introduce a new, generalized MAB formulation in which additional information on each arm may appear arbitrarily throughout the decision horizon, and study the impact of such information flows on the achievable performance and the design of efficient decision-making policies. By obtaining matching lower and upper bounds, we characterize the (regret) complexity of this family of MAB problems as a function of the information flows. We introduce an adaptive exploration policy that, without any prior knowledge of the information arrival process, attains the best performance (in terms of regret rate) that is achievable when the information arrival process is a priori known. Our policy uses dynamically customized virtual time indexes to endogenously control the exploration rate based on the realized information arrival process.