The U-Pb system is a prime geochronometer, mainly due to its occurrence as a pair of isotopically distinct but chemically identical decay systems with 235U decaying to 207Pb and 238U to 206Pb, respectively. In addition, U has suitable half-lives and is hosted in some very convenient minerals, such as the widespread and robust mineral zircon. These twin decay systems, running at different speeds, allow an immediate verification of the validity of their ages, which must be concordant to be considered valid, although under favorable circumstances, discordant data can be extrapolated to the correct age. In detail, the degree of discordance can vary greatly, and discordance can have many different causes. The progress in developments of the U-Pb method is both a history of technical discoveries and advances, as well as a history of a long struggle toward concordance, toward an understanding of the causes of discordance, and toward ways to eliminate it. Despite the enormous progress achieved in this field, the problems of U-Pb discordance have not yet been completely resolved and will be one of the main hurdles to overcome in the future. This paper reviews some of the main stages in the evolution of the method, focusing especially on the style, causes, and implications of U-Pb discordance in modern geochronology.