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The predict-then-optimize framework arises in a wide variety of applications where the unknown cost coefficients of an optimization problem are first predicted based on contextual features and then used to solve the problem. In this work, we extend the predict-then-optimize framework to a multi-task setting: contextual features must be used to predict cost coefficients of multiple optimization problems, possibly with different feasible regions, simultaneously. For instance, in a vehicle dispatch/routing application, features such as time-of-day, traffic, and weather must be used to predict travel times on the edges of a road network for multiple traveling salesperson problems that span different target locations and multiple s-t shortest path problems with different source-target pairs. We propose a set of methods for this setting, with the most sophisticated one drawing on advances in multi-task deep learning that enable information sharing between tasks for improved learning, particularly in the small-data regime. Our experiments demonstrate that multi-task predict-then-optimize methods provide good tradeoffs in performance among different tasks, particularly with less training data and more tasks.