The performance of the Planck instruments in space is enabled by their low operating temperatures, 20 K for LFI and 0.1 K for HFI, achieved through a combination of passive radiative cooling and three active mechanical coolers. The scientific requirement for very broad frequency coverage led to two detector technologies with widely different temperature and cooling needs. Active coolers could satisfy these needs; a helium cryostat, as used by previous cryogenic space missions (IRAS, COBE, ISO, Spitzer, AKARI), could not. Radiative cooling is provided by three V-groove radiators and a large telescope baffle. The active coolers are a hydrogen sorption cooler (<20 K), a 4He Joule-Thomson cooler (4.7 K), and a 3He-4He dilution cooler (1.4 K and 0.1 K). The flight system was at ambient temperature at launch and cooled in space to operating conditions. The HFI bolometer plate reached 93 mK on 3 July 2009, 50 days after launch. The solar panel always faces the Sun, shadowing the rest of Planck, andoperates at a mean temperature of 384 K. At the other end of the spacecraft, the telescope baffle operates at 42.3 K and the telescope primary mirror operates at 35.9 K. The temperatures of key parts of the instruments are stabilized by both active and passive methods. Temperature fluctuations are driven by changes in the distance from the Sun, sorption cooler cycling and fluctuations in gas-liquid flow, and fluctuations in cosmic ray flux on the dilution and bolometer plates. These fluctuations do not compromise the science data.