The "Neutrino Experiment with a Xenon TPC" (NEXT), intended to investigate the neutrinoless double beta decay using a high-pressure xenon gas TPC filled with Xe enriched in 136Xe at the Canfranc Underground Laboratory in Spain, requires ultra-low background conditions demanding an exhaustive control of material radiopurity and environmental radon levels. An extensive material screening process is underway for several years based mainly on gamma-ray spectroscopy using ultra-low background germanium detectors in Canfranc but also on mass spectrometry techniques like GDMS and ICPMS. Components from shielding, pressure vessel, electroluminescence and high voltage elements and energy and tracking readout planes have been analyzed, helping in the final design of the experiment and in the construction of the background model. The latest measurements carried out will be presented and the implication on NEXT of their results will be discussed. The commissioning of the NEW detector, as a first step towards NEXT, has started in Canfranc; in-situ measurements of airborne radon levels were taken there to optimize the system for radon mitigation and will be shown too.