To understand the history of billiards prior to the late 1700s, please see The early history of Billiards
During the 1700s, the French invented the game of Carambole. The objective was to hit both the opponents white ball and also a third (red) ball with the cue ball at a single stroke. Such a stroke was called a Carambole (later known as a 'Carom' or in England a 'Cannon'). It is presumed that the hazards (pockets) played no part in the game other than to represent annoyances to be avoided - like bunkers in golf.
By 1810, the French had started making tables without pockets at all - presumably the game was challenging enough without the hazards from the earlier game. This trend rapidly became the norm across all of Europe bar the British Isles - "The Billiard World" published by American Dudley Kavanagh in 1869, said "Here and in Spain billiards tables have four pockets, in England six and in France and the rest of Continental Europe none". And during this transition, it appears that the invention of the 'break' occurred, most likely in England but whether it happened in France, England or America, all three countries rapidly began to use the new idea in their various games.
Three cushion billiards has become the most popular form of Carom Billiards in both the USA & Western Europe - it essentially the same game as the original Carambole with the crucial difference that the cue ball must hit at least three cushions (known in the USA as rails) before striking the second ball. Obviously, this makes the game more of a geometric intrigue and much greater skill is required in order to score points.
Americans had taken to playing mainly two new forms by the late 1870's - Straight Rail and Fifteen-Ball Pool. Straight Rail was the American name for the French and European game, Carombole, played on the French Carombole table without pockets and is the forerunner of all American 'Carom Billiards' games. Conversely, American 'Fifteen-Ball Pool' or 'Sixty-one Pool' is the predecessor of all modern ''Pool'' games, being played on a table with pockets and based upon the the game of Pyramid from England, an ancestor of Snooker.
The term 'Carom' should, of course, not be confused with the much more ancient and equally popular Indian table game, Carrom or Karum. Beware also of the the ambiguity of the American usage of the term 'Billiards' - see the note at the end of the early Billiards history page for an explanation.
Balkline came next - a version of Carom Billiards with lines drawn on the table to form rectangles . When both object balls lie in the same rectangle, the number of shots that can be made is restricted. This makes the game much harder because the player must cause one of the balls to leave the rectangle, and hopefully return. In 1906 Willie Hoppe, 18, established the world supremacy of American players by beating Maurice Vignaux of France at Balkline.
Carom Billiard games, particularly Balkline, dominated the American public eye until 1919, when Ralph Greenleaf's Pool playing captured the nation's attention. Through the 1930's, both Pool and Carom Billiards, particularly three-cushion billiards, shared the spotlight.
When Balkline lost its popularity during the 1930's, Hoppe began a new career in Three-cushion billiards which he dominated until his retirement in 1952.
Carambole and its derivatives, particularly Three cushion Billiards continue to be popular throughout continental Europe and the USA while a four-ball version is very popular in some parts of Asia.
The Union Mondiale de Billard (UMB) is the controlling body for all worldwide carom games and consists of the Confédération Européenne de Billard (CEB), the Confederacion Panamericana de Billar (CPB(South America) and the Asian Carrom Billiard Federation and a few other national bodies (Algeria, Egypt, USA).
The The Billiards World Cup Association based in the Netherlands also organises global tournaments.
The Union Mondiale de Billard, is one of the primary members of the The World Confederation of Billiard Sports along with The World Snooker Federation and World Pool-Billiards Association. There is some information about it on the the BCA site.