The paper introduces a rigorous framework for quantifying coherence, identifying intuitive and computable measures of coherence. The authors adopt the viewpoint of coherence as a physical resource, defining defining conditions for measures of coherence and identifying classes of functionals that satisfy these conditions. They conclude by outlining remaining questions to complete the theory of coherence as a resource. Key points include: - Coherence plays a central role in physics, enabling applications beyond classical mechanics and ray optics. - The development of a quantitative theory of coherence follows the approach used for entanglement and reference frames. - Proper measures of coherence should satisfy monotonicity under incoherent operations and non-increasing under mixing. - The relative entropy of coherence and the $l_1$-norm of coherence are identified as valid measures, while the sum of squared off-diagonal elements violate monotonicity. - The maximally coherent state is identified as a unit for coherence, allowing for the deterministic generation of all other quantum states through incoherent operations. - The paper discusses the interconversion of coherent states and the exploitation of coherence as a resource under incoherent operations. - Future work will address the manipulation, quantification, and exploitation of coherence, including finite-dimensional and infinite-dimensional systems.The paper introduces a rigorous framework for quantifying coherence, identifying intuitive and computable measures of coherence. The authors adopt the viewpoint of coherence as a physical resource, defining defining conditions for measures of coherence and identifying classes of functionals that satisfy these conditions. They conclude by outlining remaining questions to complete the theory of coherence as a resource. Key points include: - Coherence plays a central role in physics, enabling applications beyond classical mechanics and ray optics. - The development of a quantitative theory of coherence follows the approach used for entanglement and reference frames. - Proper measures of coherence should satisfy monotonicity under incoherent operations and non-increasing under mixing. - The relative entropy of coherence and the $l_1$-norm of coherence are identified as valid measures, while the sum of squared off-diagonal elements violate monotonicity. - The maximally coherent state is identified as a unit for coherence, allowing for the deterministic generation of all other quantum states through incoherent operations. - The paper discusses the interconversion of coherent states and the exploitation of coherence as a resource under incoherent operations. - Future work will address the manipulation, quantification, and exploitation of coherence, including finite-dimensional and infinite-dimensional systems.