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In Civilization V, a unique unit (UU) is a unit that can be built only by a certain civilization in the game, replacing a generic one.
As with previous games in the series, many civilizations will have one unique unit. However, it is also common for some civilizations to have neither unique building nor improvement for an additional unique unit. For example, Rome has two units, the Legion and the Ballista, while Russia has only one, the Cossack.
Unique units usually replace other, similar units with bonuses like working better on certain terrain or a plus to combat strength. However, these unique units' bonuses are not actually too noticeable, so do not make the mistake of thinking they are of great power. The Conquistador, which is the Spanish unique replacement for Knight, will not do much better against the regular Knight (provided they have the same promotions, are on the same terrain etc.).
With the Gods and Kings expansion in 2012, unique units are no longer as unique. Allied Militaristic City-States will occasionally gift you some unique units along with normal ones. This can help you or just hinder and annoy you, depending on what they give. For example, a Mohawk Warrior has bonuses on forest and jungle, so it is obviously close to useless if you're Arabia, which usually starts out in the desert.