The Titans Tower

First Tower: The first tower was built by Cyborg's father, Dr. Silas Stone, after his son's accident. It was sited on an island in the East River, New York City. This tower was used as the headquarters for the team throughout The New Teen Titans and New Titans series. At one point, it was destroyed by Trigon, but rebuilt to Dr. Stone's design with some enhancements. It was eventually and ultimately destroyed by the Wildebeest Society in New Titans #76 (June 1991). Because the Titans had fallen out of favor with the local authorities, the tower was not rebuilt.

Second Tower: The second tower was built in the same location by Cyborg himself in the JLA/Titans miniseries. It was not an actual tower, but a hologram, with the real headquarters far beneath the ground. This was done to confuse supervillains and to keep them from attacking the real base. It was the team's base during the 1999–2002 Titans series, before being destroyed by a villain named Epsilon. This incarnation of the team disbanded before the headquarters could be rebuilt.

Third Tower: The current Titans Tower is located in San Francisco Bay. It was again designed by Cyborg, and was built by the city council, in exchange for which the Titans are responsible for dealing with the city's supervillains. This tower resembles Frank Lloyd Wright's masterpiece Fallingwater.

Clipped from marvwolfman.com:

The most notable part of this page is the first and original design I did for Titans Tower. Even before I had worked out the characterizations of the new group, I was already working on their headquarters. I wanted something splashy and distinctive and created the double-T design - one inside the other - for Teen Titans. The Titans were supposed to be given magnetic calling cards allowing them to enter the Tower, but I dropped that idea quickly. The Tower is listed as ten stories high, with a jet-copter heliport on top as well as an elevator. I obviously intended to add in something else on the right but never did. Right from the beginning I knew Titans Tower was "situated on an island in the East River."

Fun Fact: Unrealized Super Powers Titans Tower Toy Set

Fun Page: New Teen Titans V2 Annual 4