The MRI community is using quantitative mapping techniques to complement qualitative imaging. For quantitative imaging to reach its full potential, it is necessary to analyze measurements across systems and longitudinally. Clinical use of quantitative imaging can be facilitated through adoption and use of a standard system phantom, a calibration/standard reference object, to assess the performance of an MRI machine. The International Society of Magnetic Resonance in Medicine AdHoc Committee on Standards for Quantitative Magnetic Resonance was established in February 2007 to facilitate the expansion of MRI as a mainstream modality for multi‐institutional measurements, including, among other things, multicenter trials. The goal of the Standards for Quantitative Magnetic Resonance committee was to provide a framework to ensure that quantitative measures derived from MR data are comparable over time, between subjects, between sites, and between vendors. This paper, written by members of the Standards for Quantitative Magnetic Resonance committee, reviews standardization attempts and then details the need, requirements, and implementation plan for a standard system phantom for quantitative MRI. In addition, application‐specific phantoms and implementation of quantitative MRI are reviewed. Magn Reson Med 79:48–61, 2018. © 2017 International Society for Magnetic Resonance in Medicine.