Photo: Diego Delso, CC BY-SA 3.0 via Wikimedia Commons
The colonial church, built by Spaniards immediately after destroying the temple in their conquest of Aztec lands, was constructed using stones "borrowed" from the temple itself. Despite its weathered appearance, the interior is well-maintained and should still hold regular Mass, although doors may not always be open to the public.
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Tlatelolco Archaeological Site
The temple, like Tenochtitlan's Templo Mayor, was built in several layers and is now the site of continuing archaeological exploration; it occupies the largest amount of area, on the north and west side of the plaza. If you're not around for the tour, you can guide yourself along the path (complete with English plaques) that takes you through the ruins, which deposits you in front of the Church of Santiago, on the east side of the plaza.
To the south, you'll see the modern-day tower and its adjacent buildings, which were built for the Secretary of External Relations (SRE), now headquartered across the street (though they still have offices in the church's adjoining cloister). UNAM runs the building as a conference hall and cultural center, and has a few exhibits open to the public: Memorial 68 (see Museums, below) and the Blaisten Collection, showcasing modern art.
In UNAM's University Cultural Center Tlatelolco (CCUT) at the Plaza de las Tres Culturas (see Landmarks, above), Memorial 68 is a well-put together museum chronicling the events surrounding the police massacre of student protesters at Tlatelolco in 1968, around the time of the Summer Olympics, which were hosted in Mexico City that year. It's definitely worth a look, but all explanatory text and video interviews are in Spanish only, so brush up your vocabulary or bring a translator.
Antiguo Palacio de la Inquisición / Palace of the Inquisition
This museum managed by the Medicine Faculty of the National University shows the history of health care in Mexico. Features a great replica of an old pharmacy and reconstructed labs of influential medical researchers. Signage is in Spanish only. It is housed in the Inquisition Palace, built between 1732 and 1737, which was the head office of the Holy Inquisition, the religious authority famous for torturing heretic people.
Architecturally unique and utterly fascinating restored neoclassical mansion from the Porfiriato era of the 19th century with period furnishings. Guided tours (in Spanish).
Megabiblioteca
Next to Buenavista train station, it is the largest library in Mexico and functions as a central library for other libraries in the country. It is housed in an impressive modern building designed by architects Alberto Kalach and Juan Palomar. It is a must for visitors with an interest in architecture, not only for its stunning exterior design, but for its famous "floating stacks" that create the illusion of book shelves hanging from the ceiling. The library also houses a large contemporary art collection. The ground floor houses free temporary exhibits and the grounds include a botanical garden with native plants.