zurückweisen
to reject
To refuse or not accept something, for example an application, an offer, or a request.
Das Gericht hat die Klage aus formellen Gründen zurückgewiesen.
The court rejected the lawsuit on formal grounds.

to reject / turn away
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to reject
To refuse or not accept something, for example an application, an offer, or a request.
Das Gericht hat die Klage aus formellen Gründen zurückgewiesen.
The court rejected the lawsuit on formal grounds.

'Zurückweisen' (to reject / turn away) is a separable verb. In a main clause, the prefix 'zurück' goes at the end of the sentence. In subordinate clauses or in the infinitive with 'zu', the verb stays together.
Der Türsteher wies den Gast zurück.
The bouncer turned the guest away.
The verb consists of 'zurück' (back) and 'weisen' (to direct, to show). So the literal meaning is to send something or someone 'back in the direction' they came from. This helps explain all the meanings involving rejection.
Sie wies seine Hilfe höflich, aber bestimmt zurück.
She politely but firmly rejected his help.
The object of the sentence reveals the exact meaning: if you 'zurückweisen' (reject) a thing, e.g. an application, it means 'to refuse'. If you 'zurückweisen' (reject) a claim, e.g. an accusation, it means 'to deny'. If you 'zurückweisen' (turn away) a person, it means 'to send away'.
Der Antrag wurde aus formalen Gründen zurückgewiesen.
The application was rejected on formal grounds.
In the meaning 'ablehnen' (to reject/refuse), 'zurückweisen' is often used in formal, official, or legal contexts. It sounds stronger and more formal than a simple refusal.
Das Gericht wies die Berufung des Angeklagten zurück.
The court rejected the defendant's appeal.
When denying accusations or criticism, 'zurückweisen' is often intensified with adverbs such as 'entschieden' (firmly), 'scharf' (strongly), 'vehement' (vehemently), or 'kategorisch' (categorically).
Der Sprecher des Unternehmens wies die Gerüchte scharf zurück.
The company spokesperson strongly rejected the rumors.
to accept / assume
'annehmen' and 'zurückweisen' form a clear contrast here. 'annehmen' means that you accept an offer, a request, an application, or a suggestion. 'zurückweisen' means that you do not accept exactly that kind of thing and reject it. 'zurückweisen' often sounds somewhat stronger or more formal than 'ablehnen' (to decline / reject). In official or serious situations, for example with an application or a complaint, it is very appropriate. In short: someone who 'annimmt' says yes; someone who 'zurückweist' says no.

Die Firma wies den Antrag auf mehr Geld zurück.
The company rejected the request for more money.
Wir nehmen das Angebot für den Ausflug gerne an.
We are happy to accept the offer for the trip.