Circular economy concepts—including a circular bioeconomy—aim to transition the current, essentially linear, economic system to a more sustainable one. However, organizations and researchers currently define the circular economy concept differently, resulting in inconsistencies and difficulty in effectively implementing the framework. In this paper, we provide our perspective on the conceptual definitions of the circular economy, bioeconomy, and circular bioeconomy, outlining potential overlaps and differences and proposing a harmonized interpretation that stresses the importance of the carbon cycle. We conclude that the key goal of a circular economy is to slow, narrow, and close material resource loops, built on the foundation of renewable energy and non-toxic materials. Further, a sustainable bioeconomy goes beyond simply switching fossil resources with renewable, biological resources. It requires low-carbon energy inputs, sustainable supply chains, and promising disruptive conversion technologies for the sustainable transformation of renewable bioresources to high-value bio-based products, materials, and fuels. The bio-based circular carbon economy, in particular, stresses capturing atmospheric carbon via photosynthesis and exploiting this unique feature to the fullest extent possible. It sits at the intersection between the circular economy and the bioeconomy concept, resulting in a framework that focuses on closing the carbon cycle and stressing the opportunity to create an additional carbon sink capability in the technosphere by utilizing biogenic carbon for products and materials that are circulated in same or improved use cycles. Lastly, a sustainable circular bioeconomy transition will necessitate a set of consistent metrics that fit all products and industries.