
jagen
to hunt
To pursue wild animals in order to catch or kill them, usually as a sport or to obtain food.
Sein Großvater ging früher oft im Wald jagen.
His grandfather used to often go hunting in the forest.
to hunt / chase
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to hunt
To pursue wild animals in order to catch or kill them, usually as a sport or to obtain food.
Sein Großvater ging früher oft im Wald jagen.
His grandfather used to often go hunting in the forest.

to chase
To pursue someone or something quickly and eagerly in order to catch up with them or it.
Die Polizei jagte den flüchtigen Bankräuber durch die ganze Stadt.
The police chased the fleeing bank robber through the whole city.

to rush
To move somewhere very quickly, hastily, or in a hurry.
Er jagte die Treppe hoch, weil er dachte, das Telefon zu hören.
He rushed up the stairs because he thought he heard the phone.

to chase after
To strive intensely and often restlessly for something abstract, such as success, happiness, or recognition.
Sie jagte jahrelang nach beruflichem Erfolg und vergaß darüber ihr Privatleben.
For years she chased after professional success and neglected her private life because of it.
The verb 'jagen' (to hunt/chase) is the basis for the nouns 'die Jagd' (the hunt/hunting) and 'der Jäger' (the hunter). If you know the verb, you can understand the whole word family more easily.
In dieser Region ist die Jagd auf Hirsche im Herbst erlaubt.
In this region, hunting deer is allowed in the fall.
In the meaning 'verfolgen' (to pursue/chase), 'jagen' (to chase) almost always needs an accusative object. This object is the person or thing being pursued.
Die Katze jagt eine Maus durch den Garten.
The cat chases a mouse through the garden.
When 'jagen' (to rush) describes a fast movement, it is often combined with an indication of direction. It then expresses great hurry or haste.
Sie jagte aus dem Haus, um den Bus noch zu erwischen.
She rushed out of the house to still catch the bus.
When you strive for abstract things like success or happiness, 'jagen' (to chase) is always used with the preposition 'nach' (after) and the dative.
Er jagt schon sein ganzes Leben nach dem großen Geld.
He has been chasing after big money his whole life.
to follow / pursue
Both words describe running or driving after someone in order to catch up with them. Jagen usually sounds faster, more intense, and often more aggressive, while verfolgen is more neutral and can also last longer. Use jagen for a fast-paced chase and verfolgen for a more general or less intense pursuit.

Die Polizisten jagen den flüchtenden Mann mit hoher Geschwindigkeit durch die engen Straßen.
The police chase the fleeing man at high speed through the narrow streets.
Die Polizisten verfolgen den flüchtenden Mann durch mehrere Straßen.
The police pursue the fleeing man through several streets.