According to Merlin, Purgatory is an infinite, chaotic, hellish spacetime where mutilated souls and predatory monsters seek prey, the air is scorching hot and freezing cold at the same time and the earth is toxic, together destroying the flesh and bone of the living.
Time is warped in Purgatory; a minute in the real world corresponds to a year in Purgatory. It is basically a universe separated from the mortal plane. 
Through research, Merlin's father opened a portal to Purgatory and traveled through it. One real-world minute later, he returned from a year in Purgatory, looking like a haggard, grizzled invalid. He ended up dying not long after and until his dying breath muttered about the horrors of Purgatory as his daughter listened over him.
Long before the Holy War between the five clans, after losing its fight against Chaos, Cath Palug continued to commit sorrowful atrocities until the Goddess Clan deemed it too dangerous and sealed it in Purgatory. However, it later somehow managed to escape and became attached to Arthur, knowing he was chosen by Chaos itself to be its next host and waited until Arthur inherited that power so it could take it's creators power in its entirety.
At some point after the Holy War had ended, the Demon King was imprisoned in Purgatory by the Goddesses who had to sacrifice their physical forms as he, unlike the rest of the Demons, he was too powerful to be sealed away in the Coffin of Eternal Darkness.
About twenty seven-years prior to the start of the series, the Demon King kidnapped Hawk from Purgatory and used him to spy on Meliodas, given that all other creatures before were too weak and had short lifespans.
After his defeat by the Ten Commandments, Meliodas didn't move on to the Capital of the Dead but to Purgatory instead due to his curse. There, he confronts the shadowy remnants of the Demon King while waiting for his revival.
When Merlin discovers that Hawk was connected to Purgatory and that the Demon King had been observing them through Hawk's eyes, Diane suggests that they can go to Purgatory through Hawk in order to recover Meliodas's emotions that have been stolen by the Demon King. However, Merlin reveals that this would be an impossible task for them, but Ban affirms that he could make it since he is immortal. Though skeptical, Merlin uses an enchantment to send Ban to Purgatory.
In Purgatory, Ban has been transformed into a pitch black fox-like monster fighting against other beasts, but manages to regain his sense of self. Ban's hair is extremely long, showing that he has been looking for Meliodas' emotions for decades, or possibly even centuries.
His flesh is constantly destroyed and recreated through Purgatory's atmosphere and his immortality is the only reason he lasted so long, but being always on the brink of insanity. The land opens up, and he skids down a huge chasm and splatters on the ground, before coming into contact with a dragon-like creature. He fights it continuously for years on end and neither of them has fallen, yet they both know their fight will continue forever at the rate they are going.
Ban reminiscences of the others waiting for them back in the living world, Elaine especially. He asks Meliodas to hurry and come back to them, but sees the dragon is gone. He hears a voice calling for Elizabeth and looks down to see the dragon was in fact Meliodas' emotions this entire time laying unconscious next to him.
Meliodas wakes up and sees Ban to whom he believes is a hallucination but upon realizing he is the real Ban, cries over him, believing he was never going to see him, Elizabeth, or the others ever again. He explains how the Demon King had him imprisoned to make sure he would not get back to his body and eventually turned into the dragon after being consumed by rage and the Demon King decided it was okay to let him go since he could keep the exit with him to make sure Meliodas would not find it by chance.
The two of them find and kill a naturally-born creature of Purgatory that has a physical form and use its remains to create weapons and new clothing that can resist the atmosphere, and begin their search for the exit to the real world but during their voyage they are ambushed by a stranger who says he is curious how Meliodas and Ban are looking for an exit. Meliodas asks them if they know where the Demon King and the exit are, to which he agrees and says he'll show them if they take him with them, showing he is a boar who heavily resembles Hawk who is named Wild, claiming he is searching for his long-lost younger brother.
Wild leads Meliodas and Ban to his house as a hot sandstorm rolls in, and they find out that Hawk, then named Mild, was originally born in Purgatory and was taken care of by his brother until the Demon King forcefully sent him to the outside world 8 million years ago. Meliodas surmises that a minute in the outside world is equivalent to a year in Purgatory, making Hawk's appearance in Britannia 16 years ago plausible. The reason as to why Hawk was sent was to make sure that a sturdy creature was able to watch Meliodas' actions in the outside world for the Demon King, as opposed to the animals of the real world that the Demon King originally used, who ended up dying quickly. Meliodas and Ban share stories about Hawk with Wild, and Wild accepts his brother despite how Wild expected him to be more like him.
The trio resume their search for the Demon King once the storm blows over, and Meliodas tells Ban and Wild about how Zeldris became the demon he currently is but when prompted to talk about his other brother, Meliodas realizes that he cannot remember anything about Estarossa. The Demon King arrives and says that Meliodas, just like him, is unable to remember anything on his second son. Disgusted by the effeminate nature of Meliodas' emotions, the Demon King looms threateningly over the trio and questions on how Meliodas's emotions returned to his original form and why he and Ban got there, the Demon King quickly realized that Wild guided them to the door to the other side. Wild bravely charged at him, saying that he will finally force through the Demon King alongside Ban and Meliodas, but the King of Demons effortlessly flicked him away. The Demon King then began pressuring Meliodas and Ban with his sword attacks, declaring that they will never return to the living world and soon cutting off Ban's arm. After commenting on the Demon King's massive power, Ban attempted to steal the hiss power but the Demon King's innate magic power repels the assault back to Ban, instead absorbing Ban's power for himself.
He continued attacking with his sword while laughing mockingly at Ban and Meliodas for their efforts. The Demon King is soon attacked by Wild but remaining completely unfazed, he simply grabbed and threw him away while admonishing them for not giving up, confidently saying that as long as he guards the door, it is impossible for anyone to pass through the door. Not fazed by Meliodas's comments that he is not the one who decides that, the Demon King simply corrects his son that he is the one who decides everything in Purgatory and unleashes several strikes of thunder at Meliodas but Meliodas uses Full Counter against it. Remaining unfazed by it, having absorbed it with his magic power, the Demon King gives them the choice of continuing to fight and be killed or to flee, but he suddenly asks on what Meliodas knows on Estarossa, as he cannot remember him while he remembers all that happened as well as every member of the Demon Clan. When Meliodas tells him that while Estarossa was his son, he was born without the power of darkness and a weak-hearted coward who couldn't even kill a single insect, the Demon King completes that part by saying that he took pity on him and gave him a Commandment, shocking Meliodas. The Demon King then realizes that Gowther is behind this as he was the only other Commandment who betrayed him and the only one who can do such a thing.
For the next sixty years Meliodas, Ban, and Wild would fight against the Demon King for a total of 6,093 losses, until they realize the true nature of the Demon King's power is to "reverse" all magic directed at him, with attacks strengthening him while healing weakens him, allowing the three a fighting chance.
Despite getting around the Demon King's magic, the three still struggle against his raw strength and upon finding the exit, the Demon King immediately attempts to destroy Meliodas' emotions as his "real son" has no use for such useless feelings. However, he is stopped by Wild who used his ultimate ability to convert his life force into raw strength to temporarily overpower the Demon King long enough for Meliodas and Ban to escape back to the real world before the backlash catches up with him, believing Hawk will be better off with his two best friends rather than his older brother who he doesn't even remember.
Meliodas and Ban begin to fall through the portal and start getting separated, lamenting on how Wild and Hawk will never get to meet now but are interrupted by the Demon King grabbing Meliodas to prevent him from getting back to his body. In a final attempt to break free, Meliodas awaken his original magic power which destroys his father's arm due to being on the same level of power as him but realizes that by doing so, he'll be unable to exist in Britannia anymore but claims its fine and mocks his father for showing the first signs of fear as he escapes Purgatory.
With the Demon King and Cath Palug defeated, the Sins go their separate ways and Hawk plans to go to Purgatory to build a grave for Wild and runs off, but quickly realizes he doesn't know how to get to Purgatory and shouts out to Merlin for help, to which she decides to agree but says she can't help him get back out.
Hawk is immediately teleported to Purgatory and quickly realizes the hellzone it is and begins to regret his decision due to Purgatory putting his body through immense physical backlash but suddenly bumps into someone and then realizes it to be a creature who looks nearly identical to him and realizes it to be Wild, who is still alive and the brothers happily reunite but Wild claims he doesn't have much time left due to his secret technique, which is roughly about a million years and Hawk claims it to be plenty of time.
In Purgatory there are two kinds of creatures:
- Foreign species: These are the souls fallen in Purgatory that transformed by rage into pitch-black creatures with no definite form or substance. When they die they turn to dust.
- Indigenous species: These are creatures naturally born in Purgatory, with evolved bodies that withstand the extreme conditions of Purgatory. As they have "substance", when they die they leave corpses, that can be used as a source for materials to make cloths and weapons. They have an incredibly long life expectancy, exceeding millions of years. They come in different shapes and forms reminiscing different animals, including dog-like, hopper-like, pig-like and boar-like.
- Purgatory is presented as a counterpart to the Capital of the Dead; whereas the latter is a place where souls rest in peace, the Purgatory is more like a torturing prison to souls. Unlike the Capital of the Dead, which rejects the living, it seems that both living and dead can co-exist within the latter.
- Ironically, in real world religion, Purgatory is a place where people with forgiven or venial sins go, and it serves as a place of passage where souls are rehabilitated and then go to heaven and are not tortured for eternity (In fact, torture in Purgatory is a lot milder than hell)