Silicon Photomultipliers (SiPMs) are considered as a solid-state sensor alternative to photomultiplier tubes in experiments using liquid xenon (LXe) as a radiation detection medium. The main requirements are single-photon detection of the vacuum ultraviolet scintillation light from LXe at 178 nm with high resolution and detection efficiency and low noise rates. Further requirements for dark matter and double beta decay searches are ultra-low radioactivity levels of all the components including the substrates and cold electronics. Here we describe our characterisation of Hamamatsu 6×6 mm2 SiPMs in the temperature range 110–300 K in nitrogen gas, as well as long-term measurements in cold nitrogen gas at 172 K and liquid xenon at 185 K. After we introduce the experimental setups, the data acquisition schemes and analysis methods, we show the single-photon response, the gain versus bias voltage, as well as the dark and correlated noise rates. We demonstrate the long-term stability at cryogenic temperatures, and conclude that SiPM arrays are promising candidates for photosensor arrays in liquid xenon detectors. Furthermore, we study the radioactivity of the raw SiPM materials with gamma spectrometry and inductively coupled plasma mass spectrometry and conclude that SiPMs are suitable for use in low-background experiments.