
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.

to pursue / chase
To run or drive after someone, often with the intention of catching up to them or catching them.
Die Polizei verfolgte den Dieb durch die engen Gassen der Altstadt.
The police pursued the thief through the narrow alleys of the old town.

to pursue
To pursue a goal, plan, or intention consistently and purposefully.
Sie verfolgt das Ziel, die beste in ihrem Fach zu werden.
She is pursuing the goal of becoming the best in her field.

to persecute
To systematically oppress, discriminate against, or threaten someone because of their political or religious beliefs, their race, or for other reasons.
In vielen Ländern werden Journalisten verfolgt, die kritisch über die Regierung berichten.
In many countries, journalists who report critically on the government are persecuted.

to haunt
In a figurative sense: to occupy someone constantly as a thought, feeling, or memory and not let go.
Das schreckliche Bild des Unfalls verfolgt ihn bis heute in seinen Träumen.
The terrible image of the accident still haunts him in his dreams today.

to prosecute
In a legal sense: to bring a criminal offense to trial and have it punished by a court.
Die Staatsanwaltschaft wird den Betrugsfall strafrechtlich verfolgen.
The public prosecutor's office will prosecute the fraud case.
'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
In a discussion, both verbs can fit. Anyone who can follow it understands the line of thought; anyone who verfolgt it directs their attention to it for a longer period, without that automatically expressing complete understanding. Use folgen for understanding and verfolgen for attentively observing how the discussion unfolds.

Lea verfolgte die Diskussion bis zum Ende aufmerksam.
Lea followed the discussion attentively until the end.
Trotz des schwierigen Themas konnte Lea der Diskussion gut folgen.
Despite the difficult topic, Lea was able to follow the discussion well.