Third Doctor's regeneration

The Third Doctor underwent his regeneration into the Fourth Doctor as a result of radiation poisoning upon prolonged exposure to the Great One's highly unstable web of Metebelis crystals. (TV: Planet of the Spiders) He subsequently spent approximately a decade drifting through the Time Vortex, his body dying of the radiation (PROSE: Love and War), until the TARDIS brought him "home" to his laboratory at UNIT. This was the third regeneration of the 12-regeneration life cycle beginning with the First Doctor. (TV: The Time of the Doctor)

A day to come
The Third Doctor briefly began to regenerate after battling a mind parasite known as the Nurazh. The Nurazh attempted to take control of his mind, but was overwhelmed by both the Third and Fourth Doctor's minds co-existing. The parasite parished, healing the Doctor and reverting the regeneration. Jo Grant witnessed this, and described to the Doctor his next incarnation. (PROSE: The Touch of the Nurazh)

Interference
Due to interference caused by Faction Paradox, the circumstances of the Third Doctor's regeneration were altered, so that instead of regenerating on Metebelis III, he regenerated on Dust, although Mother Mathara believed there would be no substantial difference in the timeline until the Eighth Doctor. (PROSE: Interference - Book Two) The Eighth Doctor eventually rectified this. (PROSE: The Ancestor Cell)

Legacy
Reunited with the Third Doctor in the Death Zone, Sarah Jane Smith reminded him of when he changed into what he knew as "teeth and curls", only for him to explain that it was yet to happen for him. (TV: The Five Doctors)

Behind the scenes
An alternative regeneration for the Third Doctor occurred in the stageplay Doctor Who and the Daleks in Seven Keys to Doomsday, which premiered after Planet of the Spiders and directly before TV: Robot. As both Jon Pertwee and Tom Baker were not available for the production, Trevor Martin was cast as a new incarnation of the Doctor. The play began with Martin wearing a wig and costume indicating that he was the Third Doctor, not unlike Seventh Doctor actor Sylvester McCoy's portrayal of the Sixth Doctor in Time and the Rani. He would then collapse on stage, and two "audience members" would rush from their sets to help him. An image on a rear-projected screen would then show Pertwee's face turning into Martin's, indicating that he was the Fourth Doctor. As the lights turned back on, Martin would be wearing a new costume with his natural hair. As he awoke, it was revealed that the audience members were his new companions, Jenny and Jimmy, who had been sitting in the audience as a trick of immersion.