MRI has been an essential diagnostic tool in healthcare for several decades. It offers unique insights into most tissues without the need for ionising radiation. Historically, MRI has been predominantly used qualitatively, images are formed to allow visual discrimination of tissues types and pathologies, rather than providing quantitative measurements. Increasingly, quantitative MRI (qMRI) is also finding clinical application, where images provide the basis for physical measurements of, e.g. tissue volume measures and represent aspects of tissue composition and microstructure. This article reviews some common current research and clinical applications of qMRI from the perspective of measurement science. qMRI not only offers additional information for radiologists, but also the opportunity for improved harmonisation and calibration between scanners and as such it is well-suited to large-scale investigations such as clinical trials and longitudinal studies. Realising these benefits, however, presents a new kind of technical challenge to MRI practioners. When measuring a parameter quantitatively, it is crucial that the reliability and reproducibility of the technique are well understood. Strictly speaking, a numerical result of a measurement is meaningless unless it is accompanied by a description of the associated measurement uncertainty. It is therefore necessary to produce not just estimates of physical properties in a quantitative image, but also their associated uncertainties. As the process of determining a physical property from the raw MR signal is complicated and multistep, estimation of uncertainty is challenging and there are many aspects of the MRI process that require validation. With the clinical implementation of qMRI techniques and its continued expansion, there is a clear and urgent need for metrology in this field.