montieren
to assemble
To put something together or assemble it from individual parts, for example a piece of furniture or a machine.
Am Wochenende habe ich endlich das neue Bücherregal montiert.
At the weekend, I finally assembled the new bookshelf.

to assemble / mount / edit
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to assemble
To put something together or assemble it from individual parts, for example a piece of furniture or a machine.
Am Wochenende habe ich endlich das neue Bücherregal montiert.
At the weekend, I finally assembled the new bookshelf.

Verbs that end in '-ieren' are almost always regular. The past participle is formed without 'ge-' ('montiert'), and the simple past ends in '-ierte' ('montierte').
Er hat gestern den ganzen Tag Möbel montiert.
He spent the whole day assembling furniture yesterday.
In the meaning 'zusammenbauen' (to assemble), 'montieren' is typically used with objects that consist of parts. These include: furniture, a wardrobe/cabinet, a bicycle, or a machine.
Kannst du mir helfen, dieses komplizierte Gerät zu montieren?
Can you help me assemble this complicated device?
When 'montieren' means 'anbringen' (to attach/mount), the location is often given with the preposition 'an' + dative. For example: to mount 'etwas an der Wand' (something on the wall) or 'etwas an der Decke' (something on the ceiling).
Wir müssen das neue Regal sicher an der Wand montieren.
We need to mount the new shelf securely on the wall.
When film, video, or sound is involved, 'montieren' does not mean 'zusammenbauen' (to assemble), but rather 'schneiden und zusammenfügen' (to cut and combine), i.e. to edit. The context of media production makes the meaning clear.
Die Regisseurin montiert die Schlüsselszene des Films gerade neu.
The director is currently re-editing the key scene of the film.
to build
'bauen' and 'montieren' can both have to do with making or putting up something. 'bauen' is more general and is especially suitable for larger structures such as houses, roads, or simple things that are created. 'montieren' means more specifically that you put individual parts together or attach something in a place. That is why you 'montieren' a shelf, a lamp, or a device, for example. Learners may confuse the words because when you 'montieren' something, something is often also being 'gebaut'. More precisely: you build the whole thing, but you assemble or attach the parts.

Nach dem Umzug montiert Paul die Lampe über dem Esstisch.
After the move, Paul mounts the lamp above the dining table.
Die Firma baut eine neue Brücke über den Fluss.
The company is building a new bridge over the river.