verfolgen
to follow
To follow a goal, person, or event attentively, for example in the media, on television, or in a discussion.
Ich verfolge die Nachrichten jeden Tag, um auf dem Laufenden zu bleiben.
I follow the news every day to stay up to date.

to follow / pursue
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to follow
To follow a goal, person, or event attentively, for example in the media, on television, or in a discussion.
Ich verfolge die Nachrichten jeden Tag, um auf dem Laufenden zu bleiben.
I follow the news every day to stay up to date.

'Verfolgen' (to follow / pursue) is an inseparable verb. The prefix 'ver-' remains firmly attached to the stem 'folgen' (to follow) in all tenses and sentence constructions.
Er verfolgte die Diskussion mit großem Interesse.
He followed the discussion with great interest.
The meaning of 'verfolgen' (to follow / pursue) depends heavily on the object. If you pursue an abstract noun (plan, goal), it means 'to strive for' or 'to pursue'. If you follow an event (news, game), it means 'to watch' or 'to follow'. If you pursue a person, it usually means 'to chase' or 'to oppress/persecute'.
Sie verfolgt ehrgeizige Ziele.
She pursues ambitious goals.
When it is about watching media or following events, 'verfolgen' (to follow) is often combined with words such as 'die Nachrichten' (the news), 'eine Sendung' (a broadcast/show), 'ein Gespräch' (a conversation), or 'eine Entwicklung' (a development).
Hast du die politische Entwicklung der letzten Wochen verfolgt?
Have you followed the political developments of the last few weeks?
In this meaning, 'verfolgen' (to pursue) is almost always combined with abstract nouns that describe a goal or intention. Typical objects are: 'einen Plan' (a plan), 'eine Strategie' (a strategy), 'eine Absicht' (an intention), or 'ein Interesse' (an interest).
Das Unternehmen verfolgt eine neue Marketingstrategie.
The company is pursuing a new marketing strategy.
When a thought or memory 'verfolgt' (haunts) someone, this feeling is the subject of the sentence and the person is the accusative object. This structure emphasizes the person's passive role.
Die Erinnerung an den Streit verfolgt sie immer noch.
The memory of the argument still haunts her.
The expression 'strafrechtlich verfolgen' (to prosecute criminally) is a fixed legal phrase. 'Strafrechtlich' (criminally / under criminal law) is an adverb here that specifies the way in which the prosecution takes place.
Diebstahl wird in Deutschland strafrechtlich verfolgt.
Theft is prosecuted under criminal law in Germany.
to follow
'folgen' and 'verfolgen' can both have to do with staying with someone or something. In movement, 'folgen' is neutral: you walk or drive behind someone. 'verfolgen' is stronger and often sounds as if you want to catch up with someone, catch them, or not let them escape. With media or events, 'verfolgen' means that you watch or follow something attentively, for example the news or a game. 'folgen' is also used on social media for subscribing to someone. Learners may confuse the words because both express closeness or attention. But 'verfolgen' is usually more active and more intense.

Am Abend verfolgt Tom das Fußballspiel im Radio.
In the evening, Tom follows the football match on the radio.
Bitte folgen Sie dem gelben Schild bis zum Ausgang.
Please follow the yellow sign to the exit.