Is Paul Atreides the Kwisatz Haderach?

Yes, Paul Atreides is the Kwisatz Haderach. In Frank Herbert's Dune he proves it when he survives the Water of Life and can look into the part of ancestral memory no Reverend Mother can face. He arrived a generation before the Bene Gesserit planned, though, and they never got the obedient superbeing they had bred for. (Dune)

What the Bene Gesserit wanted#

The Reverend Mother Gaius Helen Mohiam explains it to Paul on Caladan, after testing him with the gom jabbar. The spice trance lets a Reverend Mother reach the memories of her female ancestors, but there is a place in that memory, the male side, that she can't look into. The Kwisatz Haderach would be a man who could look there, and whose mind could bridge space and time. (Dune)

The breeding programme aimed at him through the noble houses. The plan was for Jessica to give Duke Leto a daughter, who would marry a Harkonnen heir; their son would be the one. That would also have ended the feud between the two houses, with a Bene Gesserit man on the throne. (Dune)

Born a generation early#

Lady Jessica disobeyed. She loved Leto and wanted to give him a son, so she bore Paul. The Sisterhood's plan was thrown a generation off course, and Mohiam's test on Caladan was the Sisterhood taking his measure. (Dune)

Paul had prophetic dreams even on Caladan. On Arrakis, with spice in the food and air, they broke open: the night after his father's death, in a stilltent in the desert, he saw the futures spread out before him, including a holy war fought in his name. (Dune)

"Something unexpected"#

Paul doesn't simply accept the title. In the desert he tells his mother he is not the Kwisatz Haderach but something unexpected, and he resents the Sisterhood for the gom jabbar and for letting his father fall. (Dune)

The change comes in the last part of the novel. Paul drinks the Water of Life, the poison Fremen Reverend Mothers must change inside their bodies, and falls into a trance so deep he seems dead for three weeks. Chani brings Jessica to him, and he wakes. He makes Jessica show him the place where Reverend Mothers can't look, and he can see it. His sight now reaches the present as well as the future: he sees the Emperor's and the Guild's ships gathering over Arrakis. Before the final battle he tells Jessica plainly that he is the Kwisatz Haderach. (Dune)

At the end of Dune, Mohiam learns the cost. The being ninety generations were bred to produce will never serve the Sisterhood. Paul takes the throne by claiming Princess Irulan, and he holds the Guild and the Bene Gesserit in check through his power over the spice. (Dune)

In the films#

Villeneuve's films keep the idea and say less about it. In Dune: Part One, Mohiam tests Paul with the gom jabbar and the Bene Gesserit watch him as their possible Kwisatz Haderach, and his visions sharpen in the stilltent. In Dune: Part Two he drinks the Water of Life in the south and wakes seeing past and future, including the truth that Jessica is the Baron's daughter. (Dune: Part One; Dune: Part Two)

David Lynch's 1984 film is more direct. After Paul drinks the Water of Life and beats the Emperor, rain falls on Arrakis, and Alia declares that only the Kwisatz Haderach could do this. That rain is not in the novel. (Dune (1984 film))

A Kwisatz Haderach no one wanted#

Being the Kwisatz Haderach traps Paul. His visions show him the jihad, and he can't find a path that avoids it. In Dune Messiah he rules as a reluctant god while the war kills billions, and when he walks blind into the desert he leaves behind the Golden Path, the future he saw and refused. (Dune; Dune Messiah)

His son Leto II takes it on. Leto is a Kwisatz Haderach too, and he goes further than his father: he takes over the breeding programme himself and rules for some 3,500 years. Dune Messiah also reveals that the Bene Tleilax had made a Kwisatz Haderach of their own, which Mohiam takes to have killed itself. (Children of Dune; God Emperor of Dune; Dune Messiah)

Our reading is that this is Herbert's point. The Sisterhood got exactly what it bred for, and it did them no good, because a man who can see the future can't be ruled by people who can't. For Paul's other titles, see Paul Atreides' names and titles, and for his abilities, Paul Atreides' powers. (Dune)