
herüber
over here / across
Expresses movement from another place or another side toward the speaker or the place being considered.
Komm doch mal zu uns auf die Terrasse herüber!
Come over to us on the terrace for a moment!
over here / across
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over here / across
Expresses movement from another place or another side toward the speaker or the place being considered.
Komm doch mal zu uns auf die Terrasse herüber!
Come over to us on the terrace for a moment!

throughout
Describes extension over a certain period of time; during the entire duration of something. This usage is rather rare or regional.
Den ganzen Sommer herüber hat es fast täglich geregnet.
It rained almost every day throughout the whole summer.
The adverb 'herüber' (over here/across) consists of two parts: 'her-' and 'über'. The prefix 'her-' always expresses movement toward the speaker, while 'über' describes movement over a boundary or distance.
Der Ball rollte vom anderen Spielfeld zu uns herüber.
The ball rolled over to us from the other playing field.
'herüber' (over here/across) is very often combined with verbs of movement and then forms a separable verb. The prefix 'herüber-' comes at the end in a main clause.
Meine Nachbarin winkte mir vom Balkon herüber.
My neighbor waved over to me from the balcony.
Common verbs that are combined with 'herüber' (over here/across) are 'herüberkommen' (to come over), 'herüberschauen' (to look over / come by), 'herüberwinken' (to wave over), and 'herüberreichen' (to pass over). They all describe an action directed toward the speaker across a distance.
Schau mal, die Katze vom Nachbarn kommt zu uns in den Garten herüber.
Look, the neighbor's cat is coming over into our garden.
The use of 'herüber' (throughout) for a period of time is rare and is considered outdated or literary. In modern German, this meaning is hardly used anymore.
Das ganze Jahr herüber gab es kaum einen Tag ohne Sorgen.
Throughout the whole year, there was hardly a day without worries.
over
Rüber can colloquially describe movement toward the speaker, but the exact direction usually depends on the context. Herüber explicitly states this direction. So when someone comes from the other side toward the speaker, herüber is clearer.

Komm bitte von der anderen Straßenseite zu mir herüber.
Please come over to me from the other side of the street.
Komm bitte kurz rüber zu mir.
Please come over to me for a moment.