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.

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
'damit' and 'um ... zu' can both say what goal an action has. 'damit' is used with a subordinate clause, especially when the subordinate clause has a different subject or when you express the goal as a full clause. 'um ... zu' is used with an infinitive, usually when the same person performs the action and has the goal. The words can be confused because both name a purpose. Pay attention to the sentence form: 'damit' needs a conjugated verb, while 'um ... zu' needs an infinitive.

Mein Vater hat eine App installiert, um die Fahrzeiten im Zug zu prüfen.
My father installed an app in order to check the train times.
Er nimmt eine Jacke mit, damit ihm am Abend nicht kalt wird.
He brings a jacket so that he won't be cold in the evening.