When did the Maya build their great cities?

For the most part, the Mayas built their greatest cities on the Yucatan peninsula that in now part of Mexico. These places were tropical lowlands within what is now called tropical rain forests. When the Spanish explorers and conquistadors arrived in the early sixteenth century, the Mayas were still in control of a small area on the Northern end of the Yucatan peninsula. The Aztecs, the Tribe that first encountered Cortes' dominated the main part of what us now Mexico.