The circular economy (CE) is perceived as a sustainable economic system where the economic growth is decoupled from the resources use, through the reduction and recirculation of natural resources. In the shift towards the CE, quantifying the circularity of products and services (or their contribution to the CE) is crucial in designing policies and business strategies, and prioritizing sustainable solutions based on evidence. New circularity metrics are being developed for that purpose, but they often present contradiction in both form and content, which contributes to confusion and misunderstanding of the CE concept. This review aims to map methodological developments regarding circularity metrics for products and services, in order to: (1) identify the foundations of circularity metrics used so far and their applications, (2) evaluate the validity of current circularity metrics, based on predefined requirements and a CE definition anchored in the sustainability concept, and (3) provide recommendations on how to measure circularity. The literature search provided a wide variety of CE metrics being developed and applied (seven measurement indices, nine assessment indicators and three assessment frameworks). However, none of them are addressing the CE concept in full, potentially leading to undesirable burden shifting from reduced material consumption to increased environmental, economic or social impacts. Additionally, new metrics under-represent the complexities of multiple cycles and the consequences of material downcycling. Circularity metrics intended to sustainable decision making should be comprehensive enough to avoid burden shifting, and clearly indicate how the benefits of recycling are allocated between the primary and secondary products.