The Tigris is the eastern of the two Mesopotamian rivers, faster and less predictable than the Euphrates, and the great Assyrian cities stood on it: Ashur, Nimrud, Nineveh, and later the Parthian and Sasanian capital at Ctesiphon.
Genesis names it as one of the four rivers flowing out of Eden, alongside the Euphrates and two others whose identification is disputed, which places the biblical paradise in the Mesopotamian imagination rather than an unlocated one. That association was taken seriously for a very long time: medieval maps put Eden at the head of these rivers, and travellers looked for it there.
The river carried the cult as well as the cities. Assyrian royal inscriptions describe kings crossing it on campaign and setting up images at its source; the practice of throwing offerings into water is attested across Mesopotamia; and the flooding of both rivers underlies the flood narratives that the region produced in several languages.
Later it was a boundary between empires. Rome and Parthia, then Rome and Persia, fought repeatedly for control of the country between the rivers, and the Christian churches that grew up on the Persian side developed in Syriac and outside imperial Roman jurisdiction, which is why the Church of the East is doctrinally distinct from both Byzantium and Rome. Baghdad, founded on the Tigris in 762, became the centre of the Islamic world for five centuries.