We review research at the National Physical Laboratory (NPL) in the area of trustworthy artificial intelligence (TAI), and more specifically trustworthy machine learning (TML), in the context of metrology, the science of measurement. We describe three broad themes of TAI: technical, socio-technical and social, which play key roles in ensuring that the developed models are trustworthy and can be relied upon to make responsible decisions. From a metrology perspective we emphasise uncertainty quantification (UQ), and its importance within the framework of TAI to enhance transparency and trust in the outputs of AI systems. We then discuss three research areas within TAI that we are working on at NPL, and examine the certification of AI systems in terms of adherence to the characteristics of TAI.