Matthew was originally a man named Matthew John Cable. He first appeared in the comic, Swamp Thing. During the course of events in Swamp Thing, Matthew has a car accident and is tricked into being possessed by his wife's evil shape shifting uncle, Anton Arcane. Unwilling to put his loved ones in danger, Matthew manages to break Arcane's hold on him, only to have his injuries catch up with him. While in a coma at the hospital, corrupt hospital staff harvest some vital organs, and his wife decides to put him out of his misery and end life support. Since Matthew was in the Dreaming when he died, he became a part of the Dreaming, and was asked by Dream if he wished to be a Raven, an offer that Matthew accepted. He was surprised at how complete the transformations was, he was not a raven shaped man, but a raven to his core.
In the Sandman spin-off, The Dreaming, Matthew was accidentally pulled through to the waking world after a spell meant for the Corinthian was cast by a vengeful Echo through Anton Arcane's guidance. Once again human, Matthew was picked up by the police who contacted his wife, only to find that he had been dead 12 years, and that his wife had married and become a mother in the interim. The reunion between them was short, however, as Arcane swiftly sought out Matthew and attempted to destroy both him and Abigail. With the help of Eve and the spirits of previous Ravens, including Lucien, the First Raven, Arcane's plot was foiled and Matthew once again returned to the Dreaming.
He died twice during The Dreaming, once through the meddling of Coyote, from which he recovered, and the second time, in Litharge, accidentally shot by Lucien during a battle. He doesn't take his death badly, and told Eve not to let Lucien "blame himself."
The role of Raven was taken over by Tethys, a female of amphibious origins, who becomes the second White Raven, after Lucien, who was the original White Raven.
Today the events of The Dreaming comics are now being considered "Just a dream" by Neil Gaiman, and Matthew is again alive, and well, and still The Raven of The Dreaming.
Tethys was just a dream.