On rare occasions in the history of science, remarkable discoveries transform human society and forever alter mankind's view of the world. Examples of such discoveries include the heliocentric theory, Newtonian physics, the germ theory of disease, quantum theory, plate tectonics and the discovery that DNA carries genetic information. The science philosopher Thomas Kuhn famously described science as long periods of normality punctuated by times of crisis, when anomalous observations culminate in revolutionary changes that replace one paradigm with another. This essay examines several transformative discoveries in the light of Kuhn's formulation. We find that each scientific revolution is unique, with disparate origins that may include puzzle solving, serendipity, inspiration, or a convergence of disparate observations. The causes of revolutionary science are varied and lack an obvious common structure. Moreover, it can be difficult to draw a clear distinction between so-called normal and revolutionary science. Revolutionary discoveries often emerge from basic science and are critically dependent on nonrevolutionary research. Revolutionary discoveries may be conceptual or technological in nature, lead to the creation of new fields, and have a lasting impact on many fields in addition to the field from which they emerge. In contrast to political revolutions, scientific revolutions do not necessarily require the destruction of the previous order. For humanity to continue to benefit from revolutionary discoveries, a broad palette of scientific inquiry with a particular emphasis on basic science should be supported.