Globular clusters (GCs) exhibit star-to-star variations in specific elements (e.g., He, C, N, O, Na, Al) that bear the hallmark of high-temperature H-burning. These abundance variations can be observed spectroscopically and also photometrically, with the appropriate choice of filters, due to the changing of spectral features within the band pass. This phenomenon is observed in nearly all of the ancient GCs, although, to date, it has not been found in any massive cluster younger than 2 Gyr. Many scenarios have been suggested to explain this phenomenon, with most invoking multiple epochs of star formation within the cluster; however, all have failed to reproduce various key observations, in particular when a global view of the GC population is taken. We review the state of current observations and outline the successes and failures of each of the main proposed models. The traditional idea of using the stellar ejecta from a first generation of stars to form a second generation of stars, while conceptually straightforward, has failed to reproduce an increasing number of observational constraints. We conclude that the puzzle of multiple populations remains unsolved, hence alternative theories are needed.