deponieren
to deposit
To leave or hand over something in a safe place for storage.
Sie können Ihre Wertsachen während des Besuchs im Schließfach deponieren.
You can deposit your valuables in the locker during your visit.

to deposit / dispose
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to deposit
To leave or hand over something in a safe place for storage.
Sie können Ihre Wertsachen während des Besuchs im Schließfach deponieren.
You can deposit your valuables in the locker during your visit.

'Deponieren' (to deposit/dispose) is a rather formal and official word. You often find it in instructions, on signs in public facilities, or in administrative texts.
Bitte deponieren Sie Ihr Gepäck in den dafür vorgesehenen Fächern.
Please deposit your luggage in the compartments provided.
You can recognize the meaning from the object: if valuables, suitcases, or keys are being 'deponiert' (deposited), it is about safe storage. If waste, trash, or chemicals are being 'deponiert' (disposed of), it is about storage at a landfill.
Der Gast deponierte seinen Zimmerschlüssel an der Rezeption.
The guest deposited his room key at reception.
In the sense of 'hinterlegen' (to deposit/leave for safekeeping), the verb is often combined with places such as 'im Safe' (in the safe), 'im Schließfach' (in the locker/safe-deposit box), or 'an der Garderobe' (at the coat check). Typical objects are 'Wertsachen' (valuables), 'Gepäck' (luggage), or 'Jacken' (jackets).
Im Museum mussten wir unsere Rucksäcke an der Garderobe deponieren.
At the museum, we had to deposit our backpacks at the coat check.
In the meaning of 'lagern' (to store), 'deponieren' (to dispose/store) refers to waste. Common combinations are 'Müll deponieren' (to dispose of trash), 'Bauschutt deponieren' (to dispose of construction rubble), or 'Giftmüll deponieren' (to dispose of toxic waste). The place is usually 'auf einer Deponie' (at a landfill).
Nach dem Hausbau musste die Familie den Bauschutt fachgerecht deponieren lassen.
After building the house, the family had to have the construction rubble disposed of properly.
to hand in / give off
'abgeben' and 'deponieren' can both mean that you bring something to a certain place and leave it there. 'abgeben' is much more common in everyday life and works well for forms, packages, keys, or assignments. 'deponieren' sounds more formal and puts more emphasis on leaving something in a safe or specific place for storage. Learners may confuse the words because in both cases something no longer stays with the person. In practice: at normal everyday drop-off points, people usually say 'abgeben'; for safe depositing or official storage, 'deponieren' is more likely.

Vor der Reise hat Maria ihre Wertsachen im Hotelsafe deponiert.
Before the trip, Maria deposited her valuables in the hotel safe.
Kannst du den Schlüssel nach der Arbeit beim Hausmeister abgeben?
Can you drop off the key with the caretaker after work?