The Cryogenic Underground Observatory for Rare Events (CUORE) is the first bolometric experiment searching for neutrinoless double-beta decay (0νββ) that has been able to reach the one-tonne mass scale. The detector, located at the LNGS in Italy, consists of an array of 988 TeO_2 crystals arranged in a compact cylindrical structure of 19 towers. CUORE began its first physics data run in 2017 at a base temperature of about 10 mK and in April 2021 released its 3rd result of the search for 0νββ, corresponding to a tonne-year of TeO_2 exposure. This is the largest amount of data ever acquired with a solid state detector and the most sensitive measurement of 0νββ in 130-Te ever conducted, with a median exclusion sensitivity of 2.8E25 yr. We find no evidence of 0νββ and set a lower bound of 2.2E25 yr at a 90% credibility interval on the 130-Te half-life for this process. In this talk, we presented the current status of CUORE search for 0νββ with the updated statistics of one t yr; we also gave an update of the CUORE background model and of the measurement of the 130-Te 2νββ decay half-life, study performed using an exposure of 300.7 kg yr.