The Cryogenic Underground Observatory for Rare Events (CUORE) is the first cryogenic experiment searching for 0 νββ decay that has been able to reach the one-tonne mass scale. The detector, located at the Laboratori Nazionali del Gran Sasso (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 3 rd result of the search for 0 νββ, 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 0 νββ decay in 130Te ever conducted. We present the current status of CUORE search for 0 νββ with the updated statistics of one tonne-yr. We finally give an update of the CUORE background model and the measurement of the 130Te 2 νββ decay half-life and decay to excited states of 130Xe , studies performed using an exposure of 300.7 kgyr.