Electron collisions with atoms, ions, and molecules have represented an important area of 'applied quantum mechanics' for more than a century. This Topical Review is the write-up of the Allis Prize Lecture given by the author at the 2016 meeting of the Division of Atomic, Molecular, and Optical Physics of the American Physical Society and the 2017 Gaseous Electronics Conference. In light of the enormous size of the field, the examples presented were selected in order to tell the story of how experimental and theoretical/numerical methods have developed over time, how fruitful collaborations between data producers (experimentalists and theorists) and data users have led to significant progress, and how the results of these studies, which were often designed for fundamental research in order to push both experiment and theory to new frontiers, continue to be highly sought after for modeling applications in a variety of fields. The impact of electron collision studies on other fields, such as photoinduced processes and quantum information, is also discussed.