überlassen
to let have
To give someone something for use, disposal, or as property, often without direct compensation.
Er hat mir sein altes Fahrrad überlassen.
He let me have his old bicycle.

to leave / to entrust
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to let have
To give someone something for use, disposal, or as property, often without direct compensation.
Er hat mir sein altes Fahrrad überlassen.
He let me have his old bicycle.

'Überlassen' (to leave/to entrust) is an inseparable verb. The prefix 'über-' is not separated from the verb and is unstressed. It remains firmly attached to the verb in all sentence forms.
Sie hat mir ihre alten Bücher überlassen.
She let me have her old books.
'Überlassen' (to leave/to entrust) is typically used with a person in the dative and a thing in the accusative. The structure is: someone (nominative) leaves someone (dative) something (accusative).
Mein Vater überlässt mir für das Wochenende sein Auto.
My father is letting me have his car for the weekend.
When you give someone responsibility for an action, the structure 'es jemandem überlassen, zu + infinitive' (to leave it to someone to + infinitive) is often used. The 'es' is a placeholder here for the following infinitive construction.
Ich überlasse es dir, zu entscheiden, wohin wir essen gehen.
I'll leave it to you to decide where we're going to eat.
The expression 'sich selbst überlassen sein' (to be left to oneself) is reflexive and is formed with the verb 'sein' (to be). The reflexive pronoun adapts to the person who is being left alone.
Nach dem Stromausfall waren die Bewohner der kleinen Insel sich selbst überlassen.
After the power outage, the residents of the small island were left to themselves.
The expression 'etwas dem Zufall überlassen' (to leave something to chance) is a fixed expression. 'Dem Zufall' is always in the dative here. You can also negate the expression to say that you plan something carefully.
Bei der Sicherheit sollten wir nichts dem Zufall überlassen.
When it comes to safety, we shouldn't leave anything to chance.
to give
'geben' and 'überlassen' can both mean that something goes to another person. 'geben' is more general and works in many situations where you hand something over or make it available. 'überlassen' emphasizes more strongly that the other person is allowed to use, keep, or decide something themselves. It often sounds as if you are stepping back and leaving control or responsibility to the other person. With things like a room, a car, or a decision, 'überlassen' is often more precise than the simple 'geben'.

Paul überlässt seiner Schwester das Auto für das Wochenende.
Paul lets his sister have the car for the weekend.
Lisa gibt ihrem Kollegen die Unterlagen für den Termin.
Lisa gives her colleague the documents for the appointment.