A boy born in the woods is left alone with just a sword. Found by a local blacksmith eventually it is discovered that the sword is a bosoma, a sword with a mind of its own that possess the bearer of the weapon. Then a mysterious group approaches the boy and the blacksmith to offer a solution.

Swords are actually possessed by demons called Busoma, which constantly want to kill and fight. Not always humanity but always to see who is worthy of being possessed. I don't know, it's not clear on how they are created aside from some wizard that tries to beat death and makes a deal with the Devil to give him the knowledge to smith the most powerful weapon - in exchange for his soul. But the deal goes sour and he reneges his offer, so the Devil comes to kill the wizard (he's not actually a wizard). But they are overwhelmed and the wizard traps the Devil in the Philosopher's Stone. The smith then forges a sword with the stone as the pommel. That's the end of that plot line until someone working for a company, maybe Shoshidai (I forgot the name because they have 4 different companies looking for the fucking thing in the first episode) finds the sword at auction.

Turns out, they're not the only ones searching for it. The Demon Swords go by the name of the demon possessing it. There are other demon-possessed swords looking for it. There's at least 3 at any given time. Shiriyu, Zsoltgewinn (pronounced zolt-gyn), Fallgon, etc. all augmenting the desire to kill within their wielder. That's how Busoma are created. It's inevitable. The only way to not become a Busoma is to catch one of the people possessed and put them to sleep so they remain a Chrysalis. But this only really suspends the transformation. And there are, again, always at least 3 other Busomas searching for these Chrysalises to help them become stronger and to fight and kill.

So, one of the people looking for the Azoth sword (possessed by the Devil) actually steals it upon failing to buy it at auction. Azoth senses the buyers want for power and how jealously of his ex and it consumes him. He, possessed, seeks for his ex and she is literally about to get married at the altar but the wielder tries to kill the would-be couple. The attack is thwarted by the original buyer of the Demon sword, a member of Shoshidai, and he is bested. But it's okay because since they are both possessed, they can be friends. They join up and it's all good.

Shoshidai isn't clear. They claim to want to try to save humanity but it isn't really explained why they seek the Demon Swords and the nature of their research. The entire plot of the Shoshidai is intentionally left to the shadows. They're basically giving off mad Shinra vibes. Totally not an evil megacorp ran by billionaires seeking power or something. Zero foreshadowing.