Computed tomography was one of the first imaging modalities to require a computerized solution of an inverse problem to produce a useful image from the data acquired by the sensor hardware. The computerized solutions, which are known as image reconstruction algorithms, have thus been a critical component of every CT scanner ever sold. We review the history of commercially deployed CT reconstruction algorithms and consider the forces that led, at various points, both to innovation and to convergence around certain broadly useful algorithms. The forces include the emergence of new hardware capabilities, competitive pressures, the availability of computational power, and regulatory considerations. We consider four major historical periods and turning points. The original EMI scanner was developed with an iterative reconstruction algorithm, but an explosion of innovation coupled with rediscovery of an older literature led to the development of alternative algorithms throughout the early 1970s. Most CT vendors quickly converged on the use of the filtered back-projection (FBP) algorithm, albeit layered with a variety of proprietary corrections in both projection data and image domains to improve image quality. Innovations such as helical scanning and multi-row detectors were both enabled by and drove the development of additional applications of FBP in the 1990s and 2000s. Finally, the last two decades have seen a return of iterative reconstruction and the introduction of artificial intelligence approaches that benefit from increased computational power to reduce radiation dose and improve image quality.