The Master is a renegade Time Lord, who studied, with the Doctor, under Borusa, at the Time Lord Academy (apparently getting better grades than the Doctor), where he also met the Rani and, presumably, Drax. 

He gloried in chaos and destruction, and sought ultimate control of the universe. Like all Time Lords, he was partly telepathic, and had also developed incredible hypnotic powers. 

His favorite weapon was his Tissue Compression Eliminator.  His TARDIS was a more advanced model than the Doctor’s, with an operable chameleon circuit. 

Having led a dangerous life, the Master was already in his twelfth incarnation when he escaped from Gallifrey and arrived on Earth, at Rossini’s International Circus, his TARDIS taking the shape of a horsebox.  Under the alias of Colonel Masters, the Master took over the Farrel plastics factory and helped the Nestene to launch a

second invasion, but ended up helping the Third Doctor to banish them (EEE).  Under the alias of Prof. Emil Keller, the Master brought a Mind Parasite to Earth and tried to use it to sabotage a World Peace Conference (FFF).

The Master then became Axos’ prisoner, whom he brought to Earth.  He and the Doctor were eventually forced to work together to imprison Axos in a time loop (GGG). 

The Master stole the location of the Doomsday Machine from the Matrix and impersonated an Earth Adjudicator in order to misappropriate it, but the Doctor convinced its Guardian to destroy it (HHH).

Under the alias of Mr. Magister, the Master posed as a vicar, and the leader of a black magic coven, in Devil’s End in order to summon the Daemon Azal back to life, and to ask to be granted his power.  He failed and was arrested by UNIT (JJJ). While in prison, the Master found a way to trick his warden, Colonel Trenchard, and ally himself with the Sea Devils. After they were defeated by the Doctor, he escaped (LLL).

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Posing as Prof. Thascales,  the Master used the Kronos Crystal and TOM-TIT to summon and control Kronos.  He caused the fall of Atlantis, but was otherwise foiled by the Doctor, who interceded on his behalf with Kronos (OOO). The Master then allied himself with the Daleks, and used the Ogrons  to try to foment a war between Earth and the Draconians in the 26^th Century, but was again thwarted the Doctor (QQQ). 

Having reached his twelfth and final regeneration, the Master turned into a decaying corpse.  He was dying when Chancellor Goth found him hiding on Tersurus.  The Master promised Goth power and convinced Goth to secretly bring him to Gallifrey.  There, he helped Goth assassinate the President and frame the Fourth Doctor for the murder.  What he really wanted was access to the unlimited energy of the Eye of Harmony to boost-start a new regeneration cycle. He was thwarted by the Doctor, but stole enough energy to prolong his life and escape (4P). 
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Still a decaying husk, the Master traveled to Traken where his TARDIS assumed the shape of a Melkur, enabling the Renegade Time Lord to influence Kassia. As Melkur, the Master eventually became the new Keeper of Traken. He planned to use his control of the Source to turn the Union of Traken into an all-conquering force, and seize another body for himself.

After he was exposed and prevented from remaining Keeper by the Fourth Doctor, the Master seized and took over the body of Council Tremas (5T).  Because Tremas’s body was not that of a Time Lord, it could not regenerate, but would age instead, eventually forcing the Master to steal another body to survive. 

The Master next killed Tegan’s Aunt Vanessa, and tried to trap the  Doctor inside his TARDIS.  He followed the Doctor to Logopolis, and was responsible for its destruction, thereby endangering the whole universe. The

Master tried to use the Logopolitans’ final calculations, meant to save the Universe from collapsing, to blackmail the Universe into submission, but was thwarted by the Doctor. 

However, he caused the Doctor to fall and almost die, triggering another regeneration (5V).  The Master then kidnapped Adric and forced him to use Block Transfer Computations to create the seemingly peaceful city of Castrovalva, in reality a space-time trap for the newly-regenerated Fifth Doctor.

The Doctor discovered the true nature of Castrovalva, exposed the Master (who was disguised as Castrovalva’s Portreeve) and freed Adric. The Time Lord and his friends escaped, but the Master was almost trapped in Castrovalva’s collapse (5Z).  After escaping from Castrovalva, the Master exhausted the dynomorphic generator of his TARDIS and became stranded in the Jurassic.  He discovered the powerful alien Xeraphin, and planned to use their nucleus intelligence in his TARDIS. 
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Posing as the Arab magician Kalid, he drew a Concorde flight through a time contour in order to draw the Doctor and his TARDIS, which he needed to break down the Xeraphin’s barriers.  He then seduced the evil side of the Xeraphin and succeeded in incorporating them as part of his TARDIS.  But the Fifth Doctor secretly reprogrammed the Master’s TARDIS and exiled him to Xeriphas (6C). 

The Master later escaped from Xeriphas, taking with him the shape-changing android, Kamelion. Posing as Sir Gilles Estram, the Master forced Kamelion to assume the likeness of King John in an attempt to turn the Barons against him, and prevent the signing of the Magna Carta, but he was thwarted by the Doctor (6J).

When the Death Zone was reactivated, the High Council summoned the Master and offered him a full pardon, as well as a new cycle of regenerations, if he helped the Five Doctors.  The Master tried to seize Rassilon’s secret of immortality, but was stopped by the Brigadier.  He was eventually banished by Rassilon (6K). 
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The Master then accidentally shrank himself to doll-size with his Tissue Compression Eliminator.  He again took over Kamelion, and traveled to Sarn.  There, he plotted to bathe himself in the volcanic Numismaton Flame to restore his size and extend his powers. 

The Fifth Doctor caused him to be trapped inside the flame as it was turning caloric.  It seemed as if the Master was vaporized (6Q).  But he managed to escape and set a trap for the Sixth Doctor in 19^th Century England, where he tried to alter history by preventing the industrial revolution for his own ends. Instead, he ran afoul of the Rani, who was conducting her own biological experiments on the locals. The two renegade Time Lords entered into an uneasy alliance, but were still outwitted by the Doctor, who sent them both spinning into the outer fringes of the universe in the Rani’s TARDIS (6X). The Master eventually regained his TARDIS. 

He hired Sabalom Glitz to gain possession of the secrets of the Matrix, which had been stolen by the Andromedan Sleepers. After Glitz failed, the Master entered the Matrix. Worrying about competing with the

Valeyard in evil, he dispatched Glitz and Mel to testify at the Trial of the Doctor and expose the Valeyard. The Valeyard fled into the Matrix, pursued by the Doctor. The Master hoped the Valeyard and the Doctor would destroy each other, and tried to use Glitz to recover the Matrix’s secrets. The Doctor eventually defeated the Valeyard, but the latter had replaced the secrets with a Limbo Atrophier, which trapped the Master and Glitz inside the Matrix (7C).

The Master again managed to escape, this time seemingly without his TARDIS, but was trapped on the savage, quasi-living world of the Cheetah People. He began to turn into one of the Cheetah People, and used their teleportational abilities to escape back to Earth, where he fought the Seventh Doctor. Eventually, their conflict took them back to the dying Cheetah Planet, where the Master reverted to savagery.It is uncertain whether he managed to recover his sanity in time to escape the destruction of the Cheetah World (7P).