Europe's influence on sport is enormous. Indeed, it is difficult to think of a modern sport, apart from basketball and related sports, that does not have its origins in Europe. European sports include:

Association football (Football), which has contested origins between England and Italy. What is uncontestable is that the oldest association is The Football Association of England (1863) and the first international match was between Scotland and England (1872). It is now the world's most popular sport and is played throughout Europe.

Rugby Union and rugby league, which have similar origins to football, and whose rules in Union's case today date from 1845 (see article History of rugby union) are popular in western Europe, Romania, and Georgia, in Union's case, and in England only in League's case. The influence has been carried abroad by colonisation, with many former British colonies playing Union along with Argentina, Japan, the USA and others. Rugby League is mainly confined to New Zealand and Australia. Many other sports, such as American Football are derived from Rugby.

Athletics cannot be claimed for any continent, as its origins lie in antiquity. However the Discus, Javelin and Shot Put have their origins in ancient Greece. The Olympics, both ancient and modern, have their origins too in Europe, and have a massive influence globally.

Field Hockey as a modern game began in 18th Century England with Ireland having the oldest federation. It is popular in Western Europe, the Indian subcontinent, East Asia and the Antipodes. Ice Hockey, popular in Europe and North America derives from this sport.
Golf, one of the most popular sports in Europe, Asia and North America, has its origins in Scotland, with the oldest course being at Musselburgh.

Tennis and related games such as Table Tennis derive from the game Real Tennis which is from France. It is popular throughout the world.

Handball, which is popular in Europe and elsewhere, has its origins in antiquity. The modern game is from northern Europe with Germany having been involved in both the first women's and men's internationals.

Cycling, which is immensely popular as a means of transport has most of its sporting adherents in Europe, particularly Western Europe. The Tour de France is the world's most watched live annual sporting event. The bicycle itself is probably from France (see History of the bicycle).
Cricket, which is not so popular in Europe, being played only in northwest Europe, mainly England. It is however very popular worldwide, especially in Africa, the Antipodes and the Indian subcontinent. Sports such as Rounders and Baseball are thought to derive from this.
In addition, Europe has numerous national or regional sports which do not command a huge international following outside of emmigrant groups. These include:

Gaelic Football in Ireland, which influenced Australian rules football.
Basque Pelota in parts of Spain and France, and which has been brought to the Americas by emigrants.

Pétanque, Boules, Irish Road Bowling, Your continued donations keep Wikipedia running!
skittles, Bocce, Skittles and Bowls and others are variations of bowling games which are popular throughout western Europe and have been spread around the world