bangen um
to worry about
To have great concern or fear about the well-being or continued existence of someone or something.
Die Mutter bangt um ihr krankes Kind.
The mother is worried about her sick child.

to fear / to worry
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to worry about
To have great concern or fear about the well-being or continued existence of someone or something.
Die Mutter bangt um ihr krankes Kind.
The mother is worried about her sick child.

The verb 'bangen' (to fear / to worry) almost always needs a preposition. With 'bangen um' (+ accusative), you express concern about someone or something. With 'bangen vor' (+ dative), you express fear of a danger or event.
Die Familie bangt um den vermissten Bergsteiger.
The family is worried about the missing mountaineer.
The verb 'bangen' (to fear / to worry) belongs to the word family around 'bange' (afraid/anxious). The old expression 'Mir ist bange' means 'I am afraid' and shows the close connection.
Mir war bange, als ich allein durch den dunklen Wald ging.
I was afraid when I walked alone through the dark forest.
When you 'um etwas bangen' (worry about something), it often involves the loss of something important. Typical phrases are 'um sein Leben bangen' (to fear for one's life), 'um seinen Job bangen' (to fear for one's job), or 'um die Zukunft bangen' (to worry about the future).
Wegen der schlechten Wirtschaftslage bangen viele um ihren Arbeitsplatz.
Because of the poor economic situation, many people fear for their jobs.
When you 'vor etwas bangen' (fear something), it usually refers to upcoming unpleasant events. Common combinations are 'vor einer Prüfung bangen' (to fear an exam), 'vor einer Operation bangen' (to fear an operation), or 'vor den Konsequenzen bangen' (to fear the consequences).
Der Angeklagte bangte vor dem Urteil des Gerichts.
The defendant feared the court's verdict.
Using 'bangen' (to fear) without a preposition to express general fear is rare and belongs to an elevated, literary style.
In der Stille der Nacht begann er zu bangen.
In the silence of the night, he began to feel afraid.
to hope
'hoffen' and 'bangen' can both occur in uncertain situations. With 'hoffen', the wish for a good outcome is the main focus. With 'bangen', worry or fear is the main focus, often about a person, an outcome, or before an event. 'bangen' also sounds somewhat more serious and less everyday than 'hoffen'. Learners may confuse the words because you can hope and worry in the same situation. You hope for the good, but you 'bangt' because of the possible danger.

Heute bangt die Gruppe um ihren Auftritt, weil die Sängerin krank ist.
Today the group is worried about their performance because the singer is ill.
Jonas hofft wegen des Wetters auf ein trockenes Wochenende.
Because of the weather, Jonas is hoping for a dry weekend.