The Master, the Doctor’s oldest friend and oldest nemesis, has got out of more than a few scrapes over his many appearances in Doctor Who both on TV and in audios. The character has had almost as many appearances as the Daleks, and across all their appearances the Master has often seemingly died but then reappeared in a later story, unharmed, having escaped without even so much as regenerating.
The Master’s unique ability to escape from death is usually explained by the fact that the Master is a supervillain with a near-omnipotent ability to escape from danger so that he can be back to threaten the Doctor again in a later story, but there are some specific examples of the Master cheating death that require some more explanation.
How did the Master survive falling into the Eye of Harmony?

Eric Robert’s incarnation of the Master was trapped in the Eye of Harmony in the TV Movie, but escaped when rooms from the TARDIS were jettisoned. He later explained this to the Eighth Doctor in the Big Finish audio Day of the Master.
This question has become important recently due to Big Finish’s release of Master! starring Eric Roberts, set after the Master has escaped the Eye of Harmony and is once again at large in the universe.
How did the Master survive after running out of regenerations?

Eric Robert’s Master is merely a Human body being possessed by the Master’s consciousness, and as Day of the Master explains, he does eventually revert his withered, burnt body that he had before possessing Tremas to make Anthony Ainley’s incarnation all the way back in The Keeper of Traken.
The emaciated Master, played in this incarnation by Geoffrey Beevers, would eventually end up on the desert planet Parrak searching for a the tomb of the ancient Time Lord Artron which he believed held technology that could prolong his life or even grant him a new cycle of regenerations.Artron’s tomb did indeed contain the technology the Master was searching for, though the Master would be killed by the Ravenous before claiming his prize.
The Master survived after running out of regenerations because his future selves intervened at the last moment to use Artron’s device on their predecessors’s withered body and allow him to regenerate into a new incarnation, presumably Alex Macqueen’s.
How did the Master survive the Time War?

As we see in the Series 3 episode Utopia, the Master survived the Time War using a Chameleon Arch which rewrote his DNA to make him a Human. The Human form of the Master eventually took the name Professor Yana and was essential in building a spacecraft that transported the last of Humanity to the prophesized Utopia that existed at the end of the universe.
Unfortunately, Utopia did not exist, and Professor Yana would regain his true Time Lord self after opening his pocket watch, becoming the Master once again. He went on to murder his former assistant Chantho, who shot him in return and forced him to regenerate into his next incarnation who would then strand the Doctor and his companions at the end of the universe by stealing the TARDIS.
Big Finish expanded more on the ‘Yana’ incarnation, played by Derek Jacobi, in their War Master audio series which depict his exploits in the Time War before his eventual decision to run from the war by disguising himself as a Human.
How did the Master survive being shot by his wife?

The Master’s next incarnation, played by John Simm, would become Prime Minister of the UK and hold dominion over the entire Earth for a year until being overthrown by the Doctor and his companions and having his entire reign wiped from history.
The Master would then be shot by his wife, Lucy Saxon, and supposedly killed. His body was burned and seemingly all that survived of him was his ring. However, the Master was able to survive once again after he was resurrected by a cult of his followers.
The resurrection was botched by none other than Lucy Saxon, who sacrificed herself to cut the ritual short. This forced the Master to inhabit an unstable body that granted him bizarre powers at the cost of a constant hunger that could only be satisfied if he consumed food, and this included Humans.
How did the Master survive being stabbed by Missy?

After returning to Gallifrey and having his condition ‘cured’ by the Time Lords, the Master was stabbed by Missy after refusing to stand with the Doctor, forcing him to return to his TARDIS and regenerate into Missy. Before leaving, the Master shoots Missy in the back with his laser screwdriver, ensuring that she couldn’t stand with the Doctor either.
We see in the Big Finish audio Masterful that John Simm’s incarnation didn’t regenerate into Missy right away, however, as he would hold off his regeneration long enough to host a party and invite all his past selves with the intention of stealing their lives. Missy would intervene and disrupt the party, foiling her younger selves’ plans.
How did Missy survive on Skaro?

Missy was left in a precarious situation on Skaro at the end of The Witch’s Familiar, as she was surrounded by Daleks amid a crumbling Dalek City and seemingly had only her ‘clever idea’ to get her out of it. However, the next chronological appearance of Missy in Extremis shows her to be safe and sound.
It seems Missy survived on Skaro by either allying herself with the Daleks or escaping using a similar method to her previous successful attempts to cheat death, by teleporting away at the last second using the energy from her enemy’s weapons.
Missy does remark to the Doctor that she heard about his 24-year stay on Darillium with River Song from the Daleks, implying that she may have had dealings with them.
How did Missy survive to regenerate into Sacha Dhawan’s Master?

Although Missy seems very dead after being shot in the back by her previous incarnation at the end of The Doctor Falls, it seems as though she must have survived in order to regenerate into Sacha Dhawan’s ‘Spy’ Master who appeared in Series 12, as many officially licensed BBC sources seem to claim that this is the case.
Big Finish have already suggested that Missy survives and regenerates into a new incarnation after her death in The Doctor Falls, as Missy encounters her future self, the Lumiat, during the Missy audio series. This new female incarnation of the Master has embraced her role as a force of good in the universe, and she is able to thwart many of Missy’s schemes before being killed by Missy and presumably regenerating into Sacha Dhawan’s incarnation.
Doctor Who’s timeline is constantly in flux, however, and future revelations may change how we look at the Master’s timeline. Until then, we can only speculate as to how the Master’s story will continue.