The 58-year-old is one of Italy's most decorated managers with six Serie A titles and four domestic doubles across successful spells with Juventus and AC Milan.
Allegri arrives in Naples after his second stint with Milan was cut short in May after only one season, getting the sack following a late-season collapse that the club's American owners, RedBird Capital Partners, branded an "unequivocal failure."
A 2-1 home defeat by Cagliari condemned Milan to a fifth-place finish, meaning the seven-time European champions missed out on Champions League qualification for a second consecutive season.
Allegri's dismissal was part of a sweeping boardroom clearance at the San Siro, with senior executives Giorgio Furlani, Igli Tare, and Geoffrey Moncada also departing.
Milan had maintained a sustained challenge for the Serie A title for much of the campaign before a sharp decline in form at the end, winning just one of their final four matches.
At Napoli, Allegri replaced Antonio Conte, who announced his departure despite guiding the team to a second-place finish behind champions Inter Milan.
