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"Do you really wish to know?"
Spoilers from the books and/or games to follow!
Butcher of Blaviken is one of Geralt's aliases. Geralt gained the moniker after what transpired in the story The Lesser Evil (Polish: Mniejsze zło).

In Blaviken, Geralt slew Renfri's bandits, as well as Renfri herself, in order to save the lives of innocent villagers. However, to the general populace, it looked like Geralt attacked them for no reason as no one knew of Renfri's plans.

The Witcher (PC) Edit

This title follows the amnesiac Geralt even during the game, especially in the Temple Quarter of Vizima. Overheard in the quarter:

A townswoman: "The Butcher of Blaviken's in town. If you see him, run!!"

Also, if Geralt chooses to speak with the guards:

Guard: "I know we needed a witcher to kill the sewer beast...but the Butcher of Blaviken?"
Geralt: "What's the problem?"
Guard: "I know who you are. The city guards are watching you. I'll hear of no humans being slain in this town."
Geralt: "You'd hardly be the first to know."
Guard: "Witty...What you want?"

In the conversation with Toruviel at the Scoia'tael command post, she mentions to Geralt that he is the Butcher of Blaviken and that he "plowed Renfri before [he] killed her".

The Witcher 2: Assassins of KingsEdit

During the prologue as you enter the monastery if you strike Arthur Tailles as he rolls on the ground after he will say "Stinking murderer - I remember Blaviken"

During the Quest Troll Trouble you can intimdate one of Dmitri's comrades to reveal the current location of Dmitri by mentioning that you are the Butcher of Blaviken.

The Butcher of Blaviken is also the title of one of the achievements in the game. You obtain this achievement by killing 500 foes.

The Witcher 3: Wild Hunt Edit

When you meet the Phillip Strenger in the third game, he introduces himself by saying "blobtits call me the Bloody Baron", to which Geralt replies, "Geralt of Rivia. Blobtits call me the Butcher of Blaviken".

If Geralt chooses to spare Gaetan, he mentions his past as the "Butcher of Blaviken" and that sometimes "heads just roll", indicating that he neither absolves nor condemns the other witcher for what he did.

Gaunter O'Dimm also refers to Geralt as "Butcher of Blaviken" during their encounter in the White Orchard Inn.