The two most common pest species of rat are both native to Asia. Extensive human travel has resulted in their spread around the world. In fact, one of the common names for Rattus rattus, the black rat, is the ship rat, because they so frequently infest large ships.
The black rat is originally from India. It is well-adapted for tropical climates and is frequently driven out of more temperate environments by its cousin, the Norway rat. Rattus norvegicus, the Norway rat, is actually not native to Norway. Its origins are from Northern China. From there, it spread to Europe and eventually to the Americas. Reports of these rats stowing away on America-bound ships date to the 1770s. Both species spread from coastal shipping towns to more inland areas as time went by.