das Revier
territory
An area that an animal or group of animals claims as its habitat and defends against members of the same species.
Der Wolf markiert sein Revier mit Duftmarken.
The wolf marks its territory with scent marks.

territory / precinct
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territory
An area that an animal or group of animals claims as its habitat and defends against members of the same species.
Der Wolf markiert sein Revier mit Duftmarken.
The wolf marks its territory with scent marks.

When the topic is animals claiming an area for themselves, this meaning is always intended. Pay attention to verbs like 'verteidigen' (to defend), 'markieren' (to mark), or 'beanspruchen' (to claim).
Der Adler verteidigt sein Revier gegen andere Greifvögel.
The eagle defends its territory against other birds of prey.
In connection with the police, 'Revier' (district/precinct) can mean the area of responsibility or the police station building. The preposition helps distinguish them: 'im Revier' (in the district) vs. 'auf dem Revier' (at the police station).
Der Polizist kennt jede Straße in seinem Revier.
The police officer knows every street in his district.
'Revier' (territory) is also used generally for a person's personal area of responsibility or influence. It then describes the place or subject where someone is especially knowledgeable or in control.
Die Werkstatt ist das Revier meines Vaters, da darf niemand etwas verändern.
The workshop is my father's territory; no one is allowed to change anything there.
The use of 'Revier' for a police station is colloquial. It is a short form of 'Polizeirevier' (police precinct). In official reports, the full term is usually used.
Sie musste ihren verlorenen Ausweis auf dem Revier melden.
She had to report her lost ID at the police station.
In geographic and economic contexts, 'Revier' refers to an area with raw material deposits, especially coal. It is often combined with a geographic name, such as the 'Ruhr-Revier' (Ruhr mining district).
Das Ruhrgebiet war einst das größte Kohlerevier Europas.
The Ruhr area was once the largest coal-mining district in Europe.
district
'der Bezirk' and 'das Revier' can both refer to a defined area. 'Bezirk' is neutral and fits well with administration, city districts, or general areas. 'Revier' often has a special purpose: it can be an animal's territory, the working area of police officers or foresters, or an area in mining. Learners may confuse the words because both mean a kind of area. When talking about administration or a city, 'Bezirk' is usually correct. 'Revier' fits when someone or something is responsible for that area, works there, or defends it.

Am Morgen kontrolliert der Förster sein Revier im Wald.
In the morning, the forester checks his district in the forest.
Für den Termin ist mein Bezirk nicht zuständig, sondern der Bezirk nebenan.
My district is not responsible for the appointment; the neighboring district is.