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The phrase arms race, in its original usage, is a competition between two or more parties to have the best armed forces. Each party competes to produce larger numbers of weapons, greater armies, or superior military technology in a technological escalation. Nowadays the term is mostly used to describe any competition where there is no absolute goal, only the relative goal of staying ahead of the other competitors, essentially the goal of proving to be "better".
Examples of arms races 
From the dates 1891 to 1919, an arms race between several European countries, including Germany, France, Russia, and a few more took place. Specifically, Germany's envy of Britain's superior navy in the run up to World War I resulted in a costly building competition of Dreadnought-class ships. This tense arms race lasted until June 1914, when, after two antagonic power blocs were formed because of the rivalry, the World War broke out. After the war, a new arms race developed among the victorious Allies. The Washington Naval Treaty was only partly able to put an end to the race. Prior to WWI, a dreadnought arms race also took place in South America.
Nuclear arms race 
A nuclear arms race developed during the Cold War, an intense period between the Soviet Union and the United States. On both sides, perceived advantages of the adversary (such as the "missile gap") led to large spending on armaments and the stockpiling of vast nuclear arsenals. Proxy wars were fought all over the world (e.g. in the Middle East, Korea, Vietnam) in which the superpowers' conventional weapons were pitted against each other. After the dissolution of the Soviet Union and the end of the Cold War, tensions decreased and the nuclear arsenal of both countries were reduced.
Other uses 
More generically, the term "arms race" is used to describe any competition where there is no absolute goal, only the relative goal of staying ahead of the other competitors in rank or knowledge. An arms race may also imply futility as the competitors spend a great deal of time and money, yet end up in the same situation as if they had never started the arms race.
An evolutionary arms race is a system where two populations are evolving in order to continuously one-up members of the other population. An example of this is the escalation of drug resistance in pathogens, in step with the use of increasingly powerful drugs.
In technology, there are close analogues to the arms races between parasites and hosts, such as the arms race between computer virus writers and antivirus software writers, or spammers against Internet service providers and E-mail software writers.
See also 
- Richard J. Barnet: Der amerikanische Rüstungswahn. Rowohlt, Reinbek 1984, ISBN 3-499-11450-X (German)
- Jürgen Bruhn: Der Kalte Krieg oder: Die Totrüstung der Sowjetunion. Focus, Gießen 1995, ISBN 3-88349-434-8 (German)