Together with Pitti Palace, in 1550 the Medici bought the Boboli gardens behind the building. The name of Boboli is thought to come from the prior owner.
Nicolò Pericoli, called the "Tribolo", was called in to transform the area into one of the most spectacular Renaissance gardens. Pericoli worked at what he called his "green architecture" masterpiece until he died.
With the intervention and mannerist inventions of famous artists like Buontalenti (who created the Large Cave), Michelangelo (whose Prisons decorated the four corners of the Cave itself, before being replaced by concrete copies and transferred to the Academy Gallery) and Giambologna, Boboli became a model for all European Royal gardens, including those in the Palace of Versailles.
Besides the above mentioned Large Cave, you should visit the Amphitheatre, the Basin and the Island Tank, originally meant for the cultivation of flowers and citrus fruit trees, one of the gardens most evocative scenarios.