We present an extensive study of a key problem in online learning where the learner can opt to abstain from making a prediction, at a ccrtain cost. In the adversarial setting, we show how existing online algorithms and guarantees can be adapted to this problem. In the stochastic setting, we first point out a bias problem that limits the straightforward extension of algorithms such as ucb-n to this context. Next, we give a new algorithm, ucb- gt, that exploits historical data and time-varying feedback graphs. We show that this algorithm ben: cfits from more favorable regret guarantees than a natural extension of ucb-n. We further report the results of a series of experiments demonstrating that ucb-gt largely outperforms that extension of ucb-n, as well as other standard baselines.