So... is Katekyou Hitman Reborn! finished or not?
NO, it is not finished.
Why?
The anime finished before the manga:
Anime finished: September 25, 2010
Manga finished: November 12, 2012
MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
Watching Order (by release date)
Here you have all the anime relations we detected ordered by release date, so you can see the reason for this status in context and also use it as watching order.
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Synopsis
Tsunayoshi Sawada, nicknamed "Loser Tsuna," is an ordinary, hopelessly average middle schooler with zero prospects. His life is turned upside down by the arrival of Reborn, a baby hitman sent from Italy to train Tsuna to become the next boss of the infamous Vongola Mafia family.
Tsuna, reluctant and unwilling, is thrust into the world of the Mafia. He must overcome his shortcomings and learn to become a leader, all while navigating the everyday challenges of school and life.
With the help of Reborn's unusual training methods and a growing circle of new, and often eccentric, allies, Tsuna begins a journey of self-discovery and transformation as he prepares to inherit a powerful and dangerous legacy.