
um
at
Indicates an exact time.
Der Zug fährt um 15:30 Uhr ab.
The train departs at 3:30 p.m.
at / around / in order to
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at
Indicates an exact time.
Der Zug fährt um 15:30 Uhr ab.
The train departs at 3:30 p.m.

around
Describes movement or positioning in a circle around something.
Die Kinder saßen um das Lagerfeuer.
The children sat around the campfire.

in order to
Introduces a subordinate clause that indicates a purpose, goal, or intention.
Ich lerne Deutsch, um in Deutschland studieren zu können.
I'm learning German in order to be able to study in Germany.

around / about
Indicates an approximate time, amount, or value; circa, about.
Er wird so um den 15. Mai ankommen.
He'll arrive around May 15.

by
Indicates the extent of a change or difference.
Die Preise sind um zehn Prozent gestiegen.
Prices have risen by ten percent.

for
Expresses an exchange or something given in return.
Ich gebe dir mein altes Fahrrad um 50 Euro.
I'll give you my old bike for 50 euros.
The preposition 'um' always takes the accusative. Make sure to decline the noun and article after 'um' correctly.
Der Hund rennt um den Baum.
The dog runs around the tree.
To express a purpose or intention, you use the structure 'um ... zu' + infinitive. This is a subordinate clause in which the verb in the infinitive stands at the very end.
Er spart Geld, um ein neues Auto zu kaufen.
He's saving money in order to buy a new car.
The expression 'es geht um' + accusative is very common and means 'the topic is'. It is used to name the central topic of a film, book, or conversation.
In der Besprechung geht es um das neue Projekt.
The meeting is about the new project.
There are many verbs that are fixed with the preposition 'um'. Important examples are: 'bitten um' (to ask for), 'sich kümmern um' (to take care of), 'sich sorgen um' (to worry about), and 'sich bewerben um' (to apply for).
Könnte ich dich um einen Gefallen bitten?
Could I ask you for a favor?
'Um' on its own indicates an exact time. To express an approximate time, it is often combined with words like 'so' (about), 'gegen' (around), or 'herum' (around).
Der Film beginnt pünktlich um 20 Uhr.
The film starts promptly at 8 p.m.
To indicate the exact extent of a change, the structure '[verb] + um + [value]' is used. Typical verbs are 'steigen' (to rise), 'sinken' (to fall), 'zunehmen' (to increase), or 'reduzieren' (to reduce).
Die Temperatur ist über Nacht um fünf Grad gefallen.
The temperature fell by five degrees overnight.
so that / with it
Both damit and um ... zu can express a purpose. Damit introduces a complete subordinate clause and allows a different subject. Um ... zu is used with an infinitive and normally fits when the same person performs both actions.

Ich lese die Aufgabe noch einmal, um sie besser zu verstehen.
I am reading the task one more time in order to understand it better.
Ich erkläre die Aufgabe noch einmal, damit alle sie verstehen.
I am explaining the task one more time so that everyone understands it.