Universe

According to the Eleventh Doctor, at least his universe was born alive, but it could only become aware of itself by developing sensors across its surface, known as life forms, each of which suffered a temporary delusion of separate identity during data collection - called consciousness - but in reality had little to none individual existence. (PROSE: The Day of the Doctor) Coincidences occurred as it was a form of amusement for the universe. (TV: Closing Time) It also rewarded those, like the Doctor, who repeatedly saved it from destruction. (TV: The Snowmen)

Cosmology
Multiple universes existed (TV: Inferno) separated by an expanse called the Void. (TV: Army of Ghosts) Prior to the Last Great Time War, the Time Vortex could be used to access one another. Since the War, the pathways had closed (TV: Rise of the Cybermen) and travel usually only occurred through a Void Ship or the reality bomb breaking down the fabric of the walls of reality surrounding each universe. (TV: Army of Ghosts, The Stolen Earth / Journey's End)

Some universes existed sideways or parallel to each other. (TV: Battlefield) Likewise, some existed before (PROSE: All-Consuming Fire) or after (PROSE: Millennial Rites) each other.

Many universes developed a civilisation equivalent to that of the Time Lords - the first culture within a given reality to reach a level of sophistication which allowed them to tailor the laws of that universe to their own nature. (PROSE: Cold Fusion)

The composition of the universe consisted of galaxies which contained stars which were orbited by planetary bodies. Earth, for example, lay in the Sol system of Mutter's Spiral, the galaxy closest to the Andromeda Galaxy.

Parallels
Although it was possible for complete new universes to form by 'branching off' from specific points in another universe's history, doing so cut the lifespan of the original universe in half. (PROSE: Blood Heat)

Before
The pre-universe which existed prior to the Doctor's had a green void, stars shaped like doughnuts and physical laws closer to what might be understood as magic. (PROSE: Millennial Rites)

Knowledge of this universe was a closely guarded secret, and was only in the Doctor's possession as a result of his connection to the Matrix during his time as Lord President. (PROSE: Millennial Rites) Even then, the knowledge would ultimately have proven temporary; in his sixth and seventh lives the Doctor was fully conversant with the details and specifics of this universe (PROSE: All-Consuming Fire, PROSE: Synthespians™), but by his tenth life it was almost an article of religious faith to him that such a universe could not have existed. (TV: The Satan Pit)

This universe's equivalent to the Time Lords were the Great Old Ones, who found a means by which to survive into the next. (PROSE: All-Consuming Fire, PROSE: Millennial Rites) The Beast claimed to have existed before the universe. (TV: The Impossible Planet) It was also speculated that the Ancient Lights came from the previous universe. (TV: Secrets of the Stars)

Current
The universe of the Doctor and of the Time Lords was known as N-Space. It was in a state of constant expansion (TV: Genesis of the Daleks) and was created out of a continually existing black void roughly thirteen (TV: Secrets of the Stars) or fourteen billion years BC in what was known as Event One. (TV: Castrovalva)

The universe passed the point where it should have succumbed to heat death by the 20th century, but vented most of its entropy into E-Space. (TV: Logopolis)

The universe was replete with pocket universes and other realms or domains discrete from the wider continuum. Some of these existed in exotic special dimensions such as Calabi-Yau Space (PROSE: The Quantum Archangel), some folded into quirks of perceptual history such as the Eleven-Day Empire (PROSE: Interference - Book One) and others, such as Mictlan, had no measurable existence beyond the purely conceptual. (PROSE: Alien Bodies)

When the Olympian Prometheus stole life spores from the planet Olympus and let them go across the universe, he created life. This angered Zeus, who chained Prometheus to a mountain on an unknown planet for his actions. (COMIC: The Life Bringer!) Greyjan the Sane, on the other hand, stated that life evolved from a single ancestor cell. This didn't include more exotic lifeforms, which came into being soon after the creation of the universe. (PROSE: The Ancestor Cell)

The Time Lords of Gallifrey, as the first sentient culture to develop within their universe, (PROSE: Lucifer Rising) were responsible for setting the parameters within which the universe operated. (PROSE: Sky Pirates!) They expunged magic in favour of science as the basic governing principle. (PROSE: Cat's Cradle: Time's Crucible, Christmas on a Rational Planet) They also established the morphic field that favoured the development of humanoid life. (PROSE: Lucifer Rising) They established the Web of Time which guaranteed the universe a stable, linear history. (AUDIO: Neverland)

After the Time Lords were lost to the Last Great Time War, the universe became a more uncertain and dangerous place. Travel to parallel universes became impossible (TV: Rise of the Cybermen) and the creation of time paradoxes had a much more dangerous effect. (TV: Father's Day)

The destruction of the Doctor's TARDIS created cracks in time throughout time and space which began erasing the universe. By piloting the Pandorica (which basically contained a blueprint for the universe) into the heart of the TARDIS explosion (which was happening at all points in space and time), the Eleventh Doctor rebooted the universe, restoring it to its original state. It is unclear, however, if everything erased by the cracks were restored. Amy Pond's parents, her husband Rory and the Doctor were all restored, but this was only due to her specifically remembering them and the influence of one of the cracks on her life for years. In the case of the Doctor, she had to actively remember him in order to restore him after he was erased by the cracks while restoring the universe, so everything else erased, such as the Weeping Angels, may not have been restored. (TV: The Pandorica Opens, The Big Bang)

After
The end of the universe, also known as Event Two (PROSE: The Infinity Doctors), was variously projected to have occurred in 60 billion AD (AUDIO: Zagreus), 100 trillion AD (TV: Utopia) or, in another universe, 100,000,000,000,000,000,000,000,000,000,000 AD. (PROSE: The Infinity Doctors)

Though the Tenth Doctor denied the existence of a universe before his own (TV: The Impossible Planet), he remained aware that the death of his universe would have been followed by the birth of another. (TV: Army of Ghosts) Many time travellers killed themselves at the end of the universe in the Institute of Time because they feared the universe's heat death. (PROSE: The End)

The after-universe was a pink void interspersed with yellow mists of orbiting plankton and Saraquazel, who became its equivalent of a Time Lord. (PROSE: Millennial Rites) The Zytragupten also inhabited this universe. (COMIC: The Stockbridge Child)