The first modern multi-sport event organised were the Olympic Games, organised by the International Olympic Committee (est. 1894) for the first time in 1896 in Athens, Greece. After some badly organised celebrations (1900, 1904), the Olympics became very popular. The number of sports, initially only a few, is still growing.
At the beginning of the 20th century, another multi-sport event, the Nordic Games were first held. These Games were held in Scandinavia, and the sports conducted were winter sports such as cross country skiing and speed skating. The Nordic Games were last held in 1926, after which the 1924 Winter Sports Week in Chamonix was declared the first Olympic Winter Games.
In the 1920s, all kinds of other multi-sport events were set up. These were usually directed for a selected group of athletes, rather than everybody, which was - basically - the case with the Olympic Games. The Soviets organised the first Spartakiade in 1920, a communist alternative to the 'bourgeois' Olympic Games, and in 1922 the University Olympia was organised in Italy, the forerunner of the World University Games, meant for students only. Regional Games were another kind of multi-sport event that was established, such as the Far East Championships or the Central American Games.
The Olympic Games are still the largest multi-sport event in the world, but several others also have significance. These are:
Additional Resources
http://www.internationalgames.net/ - International Games Archive, listings of virtually all multi-sport events ever held