der Stadtbezirk
city district
A defined administrative part of a larger city, often with its own district administration.
Hamburg ist in sieben Stadtbezirke gegliedert.
Hamburg is divided into seven city districts.

city district
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city district
A defined administrative part of a larger city, often with its own district administration.
Hamburg ist in sieben Stadtbezirke gegliedert.
Hamburg is divided into seven city districts.

'Stadtbezirk' (city district) is a compound noun made up of 'Stadt' (city) and 'Bezirk' (district). So it literally refers to a district within a city.
Jeder Stadtbezirk hat eine eigene Postleitzahl.
Each city district has its own postal code.
Common phrases are 'in einem Stadtbezirk wohnen' (to live in a city district) or 'eine Stadt ist in Stadtbezirke gegliedert' (a city is divided into city districts). The word is often used with the preposition 'in'.
Berlin ist in zwölf Stadtbezirke aufgeteilt.
Berlin is divided into twelve city districts.
The term 'Stadtbezirk' (city district) belongs to administrative language. It refers to an official, administrative subdivision of a city.
Für die Anmeldung ist die Verwaltung des jeweiligen Stadtbezirks zuständig.
The administration of the respective city district is responsible for registration.
quarter / district
'das Viertel' and 'der Stadtbezirk' can both mean a part of a city. 'Viertel' sounds more natural in everyday language and often means an area with a particular character, for example 'ein ruhiges Viertel' (a quiet neighborhood) or 'ein altes Viertel' (an old neighborhood). 'Stadtbezirk' is more precise and more official. It refers to a defined part of a larger city, often with its own administration. The words can be confused because both describe parts of a city. For conversations about living somewhere or about atmosphere, 'Viertel' usually fits best. For administration, politics, or official boundaries, 'Stadtbezirk' is more appropriate.

Der Stadtbezirk plant neue Spielplätze für die Familien in der Gegend.
The city district is planning new playgrounds for the families in the area.
In diesem Viertel gibt es viele kleine Läden und Cafés.
There are lots of small shops and cafés in this neighborhood.