Religion is a brand new feature, introduced in Civilization IV.
Each religion is associated with a particluar technology, and the first civilization to discover that technology founds the religion. Upon being founded, one of your cities will be set as the "Holy City" of that religion.
Civ IV includes seven real world religions: Buddhism, Christianity, Judaism, Confucianism, Taoism, Islam, and Hinduism.

Once founded, each religion will automatically slowly spread throughout the world from their holy cities. Players can cause religions to spread faster by building missionaries and sending them out to spread the religions to other cities. You can spread religions to both your own cities and to rivals, but of course you need an Open Borders agreement to be able to get to rivals' cities. You may have multiple religions in one city. Spreading Judaism to the Buddhist city of Rome, for example, will not "convert" Rome to Judaism, but rather result in a more metropolitan city, now with both Jewish & Buddhist faithful.

Religions impact the game in a few subtle ways. At the city level, they can influence your citizens' happiness by allowing you to build religion-specific buildings such as temples, monasteries, and cathedrals. In diplomacy, civilizations with the same religions can get along better. The flip side, of course, is that civilizations of different religions will suffer in their relations. You and the computer player may also ask other civs to switch religions. And, if you control the holy city of your state religion, you will gain a line of sight bonus in all cities with your religion. The religious Civics will also affect your cities in different ways, depending on your choice.

In the end, there's really no benefit of one religion over another. Each have identical effects on game-play. The best strategy is to have as many religions as possible, and as many of the holy cities as possible.

Icon Religion Technology Shrine
Buddhism Meditation ???
Hinduism Pantheism ???
Judaism Monotheism The Temple of Solomon
Christianity Theology The Nativity
Confucianism Code of Laws ???
Taoism Philosophy ???
Islam Divine Right ???
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