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.

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
'verfolgen' and 'jagen' can both mean that someone is going after someone or something in order to catch up with them or it. 'verfolgen' is more neutral and can be used with people, vehicles, or also the police. 'jagen' sounds stronger, faster, and often more energetic. It can also feel more dramatic, as if someone is chasing after something very urgently or aggressively. If you want to say neutrally that someone is running or driving after someone, 'verfolgen' fits. If speed and eagerness are important, 'jagen' is more precise.

Der Hund jagt den Ball über die ganze Wiese.
The dog chases the ball across the whole meadow.
Die Polizei verfolgt den Wagen bis zur nächsten Kreuzung.
The police pursue the car as far as the next intersection.