In biological learning, data are used to improve performance not only on the current task, but also on previously encountered and as yet unencountered tasks. In contrast, classical machine learning starts from a blank slate, or tabula rasa, using data only for the single task at hand. While typical transfer learning algorithms can improve performance on future tasks, their performance on prior tasks degrades upon learning new tasks (called catastrophic forgetting). Many recent approaches for continual or lifelong learning have attempted to maintain performance given new tasks. But striving to avoid forgetting sets the goal unnecessarily low: the goal of lifelong learning, whether biological or artificial, should be to improve performance on all tasks (including past and future) with any new data. We propose omnidirectional transfer learning algorithms, which includes two special cases of interest: decision forests and deep networks. Our key insight is the development of the omni-voter layer, which ensembles representations learned independently on all tasks to jointly decide how to proceed on any given new data point, thereby improving performance on both past and future tasks. Our algorithms demonstrate omnidirectional transfer in a variety of simulated and real data scenarios, including tabular data, image data, spoken data, and adversarial tasks. Moreover, they do so with quasilinear space and time complexity.