As its name implies, Neo-Classicism was a kind of "new classicism". It combined musical elements from the Classical Period with the newer trends that were emerging early in the twentieth century. These classical elements included tonal centers, clarity of form, and melodic shape. To these (and many other) classical elements, neoclassicists added such modern flavorings such as quirky rhythms, spiky dissonances, and large amounts of chromaticism.
The neoclassical movement was fairly widespread, with many composers from all over Europe (and the U.S.) contributing to the sub-genre. Some of the more recognized neoclassical composers are Igor Stravinsky, Paul Hindemith, Dmitri Shostakovich, Sergei Prokofiev, and Aaron Copland, to name only a few. The motivation for the neoclassicism was simple: the heavy musical experimentation of the early part of the century left some composers wanting to "reconnect" with musical tradition. They did this, but at the same time held on to musical aspects that they had been developing during the Modern Period. The aim was not to revive old musical idioms but to simply acknowledge tradition.
While neoclassicism was a reaction against the chaotic musical period from 1910 to 1920, so too was Arnold Schoenberg's new twelve-tone method. Both tried to bring control over the previously wild music of the 20th century. The difference is that the twelve-tone method did this by creating an entirely new musical language, while neoclassicism did it by revisiting tried-and-true musical heritage.
Igor Stravinsky (perhaps the greatest neoclassicist of them all)
had this to say about the form: "The need for restriction, for
deliberately submitting to a style, has its source on the very
depths of our nature... Now all order demands restraint... In
borrowing a form already established and consecrated, the creative
artist is not in the least restricting the manifestation of his
personality. On the contrary it is more detached and stands out
better when it moves within the definite limits of a