Dream

A dream is a series of sensations which occur in the mind during sleep. Dreams could sometimes fortell the future.

The Crystal Guardian was the Guardian of Dreams. (PDA: Divided Loyalties)

The Doctor's Dreams

The Doctor dreamed of the Master seizing power with the aid of a large crystal. (DW: The Time Monster)

The Doctor dreamed of the assassination of the Lord President on Gallifrey. (DW: The Deadly Assassin)

The Doctor later dreamed that his TARDIS was invaded by a servant of the Master of the Blackhole. (TVC: Master of the Blackhole)

The Doctor dreamed that his adventures were broadcast on television. (ST: Christmas Special)

The Doctor had a dream about Sontarans in his TARDIS. (PDA: Business Unusual)

This describes the events of A Fix with Sontarans, and can be extended to include ST: Fixing a Hole.

The Doctor claimed to have once had a dream in which all his old enemies chased him around the sets of EastEnders. (NA: First Frontier)

This describes the events of Dimensions in Time, and can be extended to include Rescue, Storm in a Tikka, and Search Out Space.

The Doctor had a dream in which he had an adventure with his grandchildren John and Gillian. (DWM: The Land of Happy Endings)

Although the story implies that John and Gillian only existed in the Doctor's dreams, it's also possible that the Doctor was remembering a simpler time.

Others' Dreams

Cassandra had dreams of the future, but no one took her warnings seriously. (DW: The Myth Makers)

Leela had an apparently precognitive dream of the Fendahl. (DW: Image of the Fendahl)

Tegan Jovanka had dreams of snakes (most likely a precursor to her second encounter with the Mara). (PDA: Fear of the Dark

Ace dreamed of being chased by a Dalek wearing a swastika and chanting "Heil Doktor!" (NA: Timewyrm: Exodus)

Kadiatu Lethbridge-Stewart dreamed of the Pythia. (NA: Transit)

Ace dreamed of being naked and stabbing someone. (NA: Lucifer Rising)

Ace had dreams of a woman in red and a door she couldn't open. Bernice Summerfield dreamed of Ace strangling her while the Doctor watched, saying "it's for the good of the universe." (NA: Conundrum)

Death visited Ace in a dream. (NA: No Future)

Pain visited Benny in a dream. (NA: Set Piece)

Time Lords sometimes dreamed of being able to fly or be a different sex or have a child. (NA: Human Nature)

Benny dreamed of drinking and discussing evil with a Dalek, a Cyberman, and a Sontaran. (NA: The Also People)

Chris Cwej had dreams of being in a room with no doors. (NA: The Room With No Doors)

While on Pete's World, Rose Tyler was contacted by the Doctor, which guided her to Dålig Ulv Stranden. (DW: Doomsday)

John Smith had dreams in which he was an adventurer in time and space called the Doctor, which he recorded in his journal. (DW: Human Nature)

Luke Smith had precognitive dreams about Mrs Wormwood. (SJA: Enemy of the Bane)