Great progress has been made on sensing, perception, and signal processing over the last decades through the design of algorithms matched to the underlying physics and statistics of the task at hand. However, a host of difficult problems remain where the physics-based approach comes up short; for example, unrealistic image models stunt the performance of MRI and other computational imaging systems. Fortunately, the big data age has enabled the development of new kinds of machine learning algorithms that augment our understanding of the physics with models learned from large amounts of training data. In this thesis, we will overview three increasingly integrated physics+data algorithms for solving the kinds of inverse problems encountered in computational sensing. At the lowest level, data can be used to automatically tune the parameters of an optimization algorithm; improving its inferential and computational performance. At the next level, data can be used to learn a more realistic signal model that boosts the performance of an iterative recovery algorithm. At the highest level, data can be used to train a deep network to encapsulate the complete underlying physics of the sensing problem (i.e., not just the signal model but also the forward model that maps signals into measurements). We have shown that moving up the physics+data hierarchy increasingly exploits training data and boosts performance accordingly.