An embassy was a building where diplomats representing a specific government could live and/or work on the territory of another polity. A subdivision of an embassy was known as a consulate, while the head of an embassy was called an ambassador.
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Embassy locales
Andor
As a junior officer, Lisa Cusak worked at the Federation embassy on Andor as an attaché. While there she started a long relationship with an Andorian who worked at the Andorian Agricultural Ministry. (DS9: "The Sound of Her Voice")
Breen
In 2372, Gul Dukat suggested that if he was to get his position back as military advisor to the Detapa Council, he would post Gul Marratt to the Cardassian embassy on Breen. (DS9: "Return to Grace")
Deep Space 9
The Klingons, Bajorans, and the Federation maintained consulates, and the Vulcans an embassy, on Deep Space 9's Promenade by 2370. (Star Trek: Deep Space Nine set decoration)
Earth
In the 22nd century, the Vulcans had a consulate on Earth in San Francisco. T'Pol worked in the Vulcan consulate in San Francisco for two years before assigned to the Enterprise in 2151. (ENT: "Fusion")
By 2155, Vulcan consulates also existed in Berlin and Canberra. Terra Prime operatives saw no indication of the Vulcans leaving the consulates in those locations despite an ultimatum for all aliens to leave the Sol system. The Andorians also had an embassy on Earth that year, which was targeted by Terra Prime protesters. (ENT: "Terra Prime")
In the mid-2270s, Commander Sonak was killed in a transporter accident while beaming up to the USS Enterprise. Admiral James T. Kirk asked Starfleet Command to send his condolences to Sonak's family, who could be reached through the Vulcan embassy. (Star Trek: The Motion Picture)
The holographic wife of Kenneth, Charlene, was supposed to speak at the Bolian embassy in 2373. (VOY: "Real Life")
Farius Prime
In 2374, the Orion Syndicate mounted a failed attempt to break into the Klingon embassy on Farius Prime to assassinate their ambassador. (DS9: "Honor Among Thieves")
Khefka IV
Before being recalled to Trill, Verad had been the communications clerk of the Federation consulate on Khefka IV. (DS9: "Invasive Procedures")
Mazar
In 2152, a Vulcan consulate existed on Mazar. Mazarites opposed to corrupt government officials asked the consulate for help to expose the criminals. (ENT: "Fallen Hero")
Mordan IV
In 2364, the staff of the Federation embassy on Mordan IV was taken hostage. (TNG: "Too Short a Season")
Qo'noS

The Orion embassy outpost on Qo'noS
The Orions operated an embassy outpost on Qo'noS, which was built above an ancient shrine to Molor. It served as a center for unscrupulous activities for individuals of many species. (DIS: "Will You Take My Hand?")
One of Odo's friends in Starfleet Intelligence was formerly assigned to the Federation embassy on Qo'noS. This friend still had a contact with a former agent of Klingon Intelligence by 2371. (DS9: "Visionary")
Romulus
Early in his Obsidian Order career, Elim Garak was posted to the Cardassian embassy on Romulus. According to Garak, however, he was merely a gardener there. (DS9: "Broken Link")
Tulgana IV

The Federation Embassy on Tulgana IV in 2380
The Federation had an embassy on Tulgana IV that was visited by Ensigns Beckett Mariner and Brad Boimler, as well as the Klingon general K'orin, in 2380. (LD: "Envoys")
Vulcan

The United Earth Embassy on Vulcan in 2154
The United Earth Embassy on Vulcan was bombed in 2154. (ENT: "The Forge")