Twenty years ago, the Vongola Decimo took his seat as the head of the Vongola famiglia, which was a prominent mafia family in Italy. He was tutored by Reborn and had to endure several trials to become boss, discovering his true reason for fighting - to protect his friends and family.
Now, another student must take a similar path. She must endure the hardships imposed upon her by her tutor so that she may be prepared for the life before her as a possible successor to Tsuna, the current Vongola boss. She must also rally her friends to help her defeat innumerable enemies on her path to becoming boss. On the way, she will forge bonds that will last a lifetime, discover her true nature, and realize her purpose in life, whatever that is.
A group of students in Venice, Italy do not seem to be the most fit or likely to run a powerful mafia organization. Indeed, at the outset of their adventure, they really aren't much to speak of. But through various trials and tests which they must overcome through skill, will, and pure luck, a few kids will evolve into powerful mafiosi fit to influence the actions of many individual organizations below them.
It all starts on a sleepy Sunday in the city of water...

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It's short, I know. I will add to it later, but the gist is there.