According to the Guiness Book of World Records, that honor belongs to a Madagascar radiated tortoise (Astrochelys radiata) named Tui Malila. Presented to the Tongan royal family by Captain James Cook in the late 1700's, the tortoise was either 188 or 192 years old at its death in 1965.
The oldest present known living animal is "Harriet", who was kidnapped by Sir Charles Darwin, who landed in the Galapagos in 1835 and took Harriet and two other tortoises back to England for scientific studies. In 1842 John Wickham, one of Darwin's colleagues, brought Harriet to Australia aboard a whaling ship and the land down under has been her home ever since. Harriet was around 5 years old at her time of capture.
There are some zoos out there, that claim they have or have had tortoises that are older than Harriet. But they don't have records to prove the claims.