
träumen
to dream
To experience images, thoughts, and feelings while asleep.
Ich habe letzte Nacht geträumt, dass ich fliegen konnte.
I dreamed last night that I could fly.
to dream
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to dream
To experience images, thoughts, and feelings while asleep.
Ich habe letzte Nacht geträumt, dass ich fliegen konnte.
I dreamed last night that I could fly.

to dream of
To wish for something very strongly or imagine something in one's fantasy.
Schon als Kind hat sie davon geträumt, Astronautin zu werden.
Even as a child, she dreamed of becoming an astronaut.

to daydream
To be lost in thought and inattentive; to not be paying attention.
Hör auf zu träumen und konzentrier dich auf deine Hausaufgaben!
Stop daydreaming and concentrate on your homework!
When describing the content of a dream or a wish, you use the preposition 'von' followed by the dative. This structure is very typical for the meanings 'to experience while asleep' and 'to wish for something'.
Ich träume oft von meiner Kindheit.
I often dream of my childhood.
The meaning 'to be inattentive' is often used as a command ('Träum nicht!' — Don't daydream!). Whether a dream while asleep or a wish is meant is revealed by the context: time expressions like 'letzte Nacht' (last night) point to a sleeping dream.
Was hat der Lehrer gesagt? Ich habe gerade geträumt.
What did the teacher say? I was just daydreaming.
To describe a complex dream while asleep, a subordinate clause with 'dass' (that) is often used. The conjugated verb of the subordinate clause then comes at the end.
Sie hat geträumt, dass sie eine wichtige Prüfung verpasst.
She dreamed that she missed an important exam.
In the meaning 'to be inattentive', 'träumen' (to daydream) often has a negative or reproachful undertone. It is usually used to tell someone to concentrate.
Träumst du schon wieder? Pass bitte im Unterricht auf!
Are you daydreaming again? Please pay attention in class!
to hope
Hoffen and von etwas träumen can both express a strong desire for the future. When you hoffen, you consider fulfilling that desire possible. Someone who von etwas träumt, on the other hand, can also imagine an unrealistic future in their mind.

Ohne Geld träume ich davon, bald um die Welt zu reisen.
Without any money, I dream of traveling around the world soon.
Mit meinen Ersparnissen hoffe ich, bald um die Welt reisen zu können.
With my savings, I hope I’ll soon be able to travel around the world.