The history of the spread and use of the plants of the genus Capsicum is a curious one. These plants, native to the Americas, are now grown throughout the world, and appear in recipes that come back to America in ethnic restaurants. The genus is represented in American cuisine by all of the varieties of green peppers, chili peppers, Oriental hot peppers, pimentos, and paprika. The amazing part of this is the speed with which the plants spread and were adopted into local cultures along the routes of the Spanish and Portuguese explorers of the 15th and 16th centuries.
Here's the history of how they spread in an article written by my father.