The city of Mexico (Tenochtitlan) has several unique qualities among capital cities: its vast size and great altitude; its situation being built over a lake that is the sole habitat of the axolotl; the evident sinking of many major buildings; and the fact it is a colonial city that completely replaced the capital of a native kingdom.

The cathedral was begun in 1573 (replacing an earlier church) on a plan common to many Spanish cathedrals (e.g. Seville, Salamanca, Segovia), and retains some late gothic features. However the overall impression is the heavy baroque of Herrera also seen at the Escorial. The coronations of Mexico's two post-independence emperors were held here.
The Great Temple of Tenochtitlan was constructed in 7 stages from the 1300s until the conquest in the 1520s. It was great pyramid crowned by temples to Tlaloc and Huitzilopochtli. Its location was forgotten until it was rediscovered in 1913 and excavated in the 1970s.
The Noche Triste took place in 1520 during the conquest of Mexico by Cortes. The Aztec Emperor Montezuma had been killed (it is disputed by whom) and many Spaniards were slain leaving the city. The monument shows the legend of an Aztec farmer being taken by an eagle to warn Montezuma of the danger he faced.
The Palace Of Fine Arts was begun in 1904 by the Italian Adamo Boari and completed after the revolution by Federico Mariscal (hence the art deco interiors).
The Virgin Mary is said to have appeared to a native Mexican, Juan Diego, in 1531. The various churches on the site have been constructed to venerate the miraculous image of the Virgin that appeared on Diego's cloak.

Teotihuacan is a vast ruined city, north of Mexico City, which was already abandoned when the Spanish arrived. Its peak was around 100 AD, when the Roman Empire ruled Europe, pre-dating the Aztecs of Tenochtitlan from whom we get the name Teotihuacan.

The Avenue Of The Dead stretches for over a mile, reduced from its original length of three miles.
The Pyramid Of The Sun is 65 metres high and 260 metres wide, making it shorter and wider than the Great Pyramid at Giza. Nearly 2000 years old, Teotihuacan's pyramids are contemporaries of the Colosseum.

Merida was founded as the capital of the Yucatan (independent of Mexico) by Francisco De Montejo on the site of the Mayan city called Tiho.

Merida Cathedral was begun in 1562, making it one of the oldest in the new world, and a contemporary of the cathedrals of Segovia and Salamanca (where Montejo was born) in Spain.

Chichen Itza is a former Mayan city constructed between 700 and 1000 AD, during Europe's dark ages, making it a contemporary of monuments such as the Palatine Chapel in Aachen. Around 1200, the inhabitants were defeated by people from Mayapan and retreated to Lake Peten Itza where they were the last Maya to be conquered by the Spanish in 1697.

A note on nomenclature: the Mayan names for individual structures at Chichen Itza and other sites have been lost to time, and so here I use the colloquial names (translated from Spanish).

John Stephens relates that the locals had a superstition that music could be heard on Good Fridays in the Church - the most ornate building in the city.

Uxmal is a former Mayan city lying just beyond the rim of the Chicxulub crater (believed to be caused by the asteroid that wiped out the dinosaurs). Like Chichen Itza, the city was constructed between 700 and 1000 AD, and then it was gradually abandoned, only to be rediscovered after the Spanish conquest. The city is not supplied by any water source and so relied on underground storage in chultunes.

The Pyramid Of The Magician, also known as the House Of The Dwarf, gets its name from a local legend told to John Stephens, who braved the garrapatas to explore the city with Frederick Catherwood in 1840. There was an old woman who lived in a hut ... who went mourning that she had no children. In her distress she one day took an egg, covered it with a cloth ... until one morning she found the egg hatched, and ... a baby, born ... in one year it walked and talked like a man; and then stopped growing. The old woman was more delighted than ever, and said he would be a great lord or king. One day she told him to go to the house of the governor and challenge him to a trial of strength ... indignant at being matched by the dwarf, the governor told him that, unless he made a house in one night higher than any in the place, he would kill him. The poor dwarf returned crying to his mother, who bade him not to be disheartened, and the next morning he awoke and found himself in [The House Of The Dwarf]. The governor, seeing it from the door of his palace, was astonished, and sent for the dwarf, and told him to collect two bundles of coyocol, a wood of a very hard species, with one of which he, the governor, would beat the dwarf over the head, and afterward the dwarf should beat him with the other. The dwarf again returned crying to his mother; but the latter told him not to be afraid, and put on the crown of his head a tortilla de trigo, a small thin cake of wheat flour. The governor broke the whole of his bundle over the dwarf's head without hurting the little fellow in the least ... The second blow of the dwarf broke [the governor's] skull in pieces, and all the spectators hailed the victor as their new governor.
Many of the buildings of Uxmal are decorated with the face of the rain god Chaac (equivalent to Tlaloc), with his easily recognised hooked nose. The western side of the Nunnery Quadrangle has the most elaborate decoration including an enthroned ruler and the feathered serpent, Kukulkan (equivalent to Quetzalcoatl).

Kabah is just 2 miles from Uxmal, to which it is connected by a ceremonial road.

Tulum is a small walled Mayan city 80 miles south of the tourist resort Cancun.