The Brothers Karamazov was originally imagined as a five-volume epic entitled The Great Sinner (source).
Dostoevsky suffered from epilepsy and drew on his own experiences in his characterization of Smerdyakov (source).
Some believe that Fyodor Karamazov is based on Dostoevsky's father, Mikhail Andreevich Dostoevsky. Family legend had it that his father was murdered by the peasants of his village, Chermashnia, because he kept sleeping with their women, a legend that was never confirmed (source).
Sigmund Freud wrote a famous study of Dostoevsky's The Brothers Karamazov called "Dostoevsky and Parricide," in which he argued that Dostoevsky was working out his guilt about wishing for his father's death (source).
The TV show Seinfeld had an episode featuring the Flying Karamazov Brothers, a real acrobatic act that appeared in the episode as the Flying Sandos Brothers (source).