herum
around
Indicates movement or position in a circle around something or along a boundary.
Die Katze schleicht um den heißen Brei herum.
The cat is beating around the bush.

around / about
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around
Indicates movement or position in a circle around something or along a boundary.
Die Katze schleicht um den heißen Brei herum.
The cat is beating around the bush.

'Herum' (around/about) is made up of the directional particles 'her' (movement toward the speaker) and 'um' (around/in a circle). It often describes circular or undirected movement.
Komm mal schnell herum, ich will dir etwas zeigen.
Come around quickly; I want to show you something.
'Herum' is a very common separable prefix. In verbs like 'herumlaufen' (to run/walk around), 'herumstehen' (to stand around), or 'herumliegen' (to lie around), it often gives the meaning of aimlessness or inactivity.
Die Kinder laufen den ganzen Tag im Garten herum.
The children run around in the garden all day.
When 'herum' means 'ungefähr' (approximately/about), it always comes after the time or quantity it refers to. The expression itself is often introduced with 'um' or 'so'.
Ich bin so gegen 10 Uhr herum bei dir.
I'll be at your place around 10 o'clock or so.
The expression 'herum sein' (to be over) means that a period of time has come to an end. The subject is the period of time itself.
Endlich ist die anstrengende Woche herum!
Finally, the exhausting week is over!
The idiom 'um etwas herumkommen' (to get around something) means that you cannot avoid something unpleasant. It is usually used in the negative.
Um die Prüfung wirst du nicht herumkommen.
You won't get around the exam.
at / around / in order to
'um' and 'herum' can both have to do with movement or position around something. 'um' is used as a preposition before a noun: 'um den Tisch' (around the table). 'herum' is often used together with 'um' or after the verb, and emphasizes movement along the outside: 'um den Tisch herum' (around the table). Both can also make approximate statements, for example about a time. Learners may confuse them because they often appear together. Put simply: 'um' usually needs a noun after it, while 'herum' more strongly describes the type of movement or makes the statement looser.

Mein Vater kommt am Abend um acht Uhr herum nach Hause.
My father comes home at around eight o'clock in the evening.
Wegen der Baustelle fährt der Bus um das Zentrum.
Because of the construction site, the bus goes around the city center.