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Inspired by the demands of real-time time-series forecasting, we develop and analyze optimistic online learning algorithms under delayed feedback. We present a novel "delay as optimism" analysis that reduces online learning under delay to optimistic online learning. This reduction enables optimal regret bounds for delayed online learning and exposes how side-information or optimistic "hints" can be used to combat the effects of delay. We use these theoretical tools to develop the first optimistic online learning algorithms that require no parameter tuning and have optimal regret guarantees under delay. These algorithms --- DORM, DORM+, and AdaHedgeD --- are robust and practical choices for real-world time-series forecasting. We conclude by benchmarking our algorithms on four subseasonal climate forecasting tasks, demonstrating low regret relative to state-of-the-art forecasting models.