rundum
completely
Completely, entirely, in every respect.
Es war ein rundum gelungener Abend.
It was a completely successful evening.

completely / all around
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completely
Completely, entirely, in every respect.
Es war ein rundum gelungener Abend.
It was a completely successful evening.

'Rundum' is made up of the words 'rund' (round) and 'um' (around). This origin explains both the spatial meaning ('ringsherum', all around) and the figurative meaning ('vollständig', completely; 'in jeder Hinsicht', in every respect).
Wir haben das Haus rundum renoviert.
We renovated the house completely.
You can recognize the meaning from the context. If 'rundum' refers to an evaluation or a feeling, e.g. 'zufrieden' (satisfied) or 'gelungen' (successful), it means 'completely'. If it refers to a place, e.g. 'Haus' (house) or 'Garten' (garden), it means 'ringsherum' (all around).
Ich bin mit dem Ergebnis rundum zufrieden.
I am completely satisfied with the result.
In the meaning 'vollständig' (completely), 'rundum' often stands before positive adjectives or participles. Typical combinations are 'rundum zufrieden' (completely satisfied), 'rundum glücklich' (completely happy), or 'ein rundum gelungenes Fest' (a completely successful celebration).
Dank der guten Organisation war es ein rundum gelungenes Event.
Thanks to the good organization, it was a completely successful event.
In the spatial meaning, 'rundum' describes what surrounds something on all sides. It can stand alone or reinforce a preposition such as 'um' (around).
Von dem Turm aus hat man einen tollen Blick rundum.
From the tower, you have a great view all around.
completely
'völlig' and 'rundum' can both mean that something applies fully or completely. 'völlig' is very general and intensifies many different statements. 'rundum' is closer to 'in every respect' and fits especially when several sides or areas are meant. For example, you say 'völlig falsch' (completely wrong), but more likely 'rundum zufrieden' (completely satisfied) when everything about a situation is good. 'rundum' often sounds more positive and more comprehensive.

Nach dem Training war das Team rundum zufrieden mit dem Ergebnis.
After the training session, the team was completely satisfied with the result.
Zur Vorbereitung auf den Termin habe ich den Bericht völlig neu geschrieben.
To prepare for the appointment, I completely rewrote the report.