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The Cryogenic Underground Observatory for Rare Events (CUORE) is the first cryogenic experiment searching for$$0\nu \beta \beta $$$0\nu \beta \beta$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$${\mathrm{TeO}}_{2}$$${\mathrm{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$$3{\mathrm{rd}}$$$3\mathrm{rd}$result of the search for$$0\nu \beta \beta $$$0\nu \beta \beta$, corresponding to a tonne-year of$$\mathrm{TeO}_{2}$$${\mathrm{TeO}}_2$exposure. This is the largest amount of data ever acquired with a solid state detector and the most sensitive measurement of$$0\nu \beta \beta $$$0\nu \beta \beta$decay in$${}^{130}\mathrm{Te}$$${}^{130}\mathrm{Te}$ever conducted . We present the current status of CUORE search for$$0\nu \beta \beta $$$0\nu \beta \beta$with the updated statistics of one tonne-yr. We finally give an update of the CUORE background model and the measurement of the$${}^{130}\mathrm{Te}$$${}^{130}\mathrm{Te}$$$2\nu \beta \beta $$$2\nu \beta \beta$decay half-life and decay to excited states of$${}^{130}\mathrm{Xe}$$${}^{130}\mathrm{Xe}$, studies performed using an exposure of 300.7 kg yr.