This paper introduces a rigorous framework for quantifying coherence in quantum systems. The authors define coherence as a physical resource and identify intuitive and computable measures of coherence. They establish conditions that any valid coherence measure must satisfy, distinguishing between valid and invalid measures. The paper outlines the need for a complete theory of coherence as a resource, highlighting the importance of coherence in quantum information science and its role in biological systems. The authors define incoherent states as those diagonal in a fixed basis and incoherent operations as quantum operations that map incoherent states to incoherent states. They distinguish between two types of incoherent operations: those that do not retain measurement outcomes and those that do. A maximally coherent state is defined as a state that can be used to deterministically generate all other d-dimensional quantum states through incoherent operations. The paper presents two key coherence measures: the relative entropy of coherence and the $ l_1 $-norm of coherence. These measures satisfy the required properties of monotonicity under incoherent operations and convexity. The relative entropy of coherence is defined as the difference between the von Neumann entropy of a state and the entropy of its diagonal counterpart. The $ l_1 $-norm of coherence is the sum of the absolute values of the off-diagonal elements of a density matrix. The authors also discuss other potential coherence measures, such as those based on the $ l_2 $-norm and fidelity, but show that some of these do not satisfy the required monotonicity conditions. The paper concludes with an outlook on future research directions, including the development of a complete theory of coherence as a resource, the exploration of infinite-dimensional systems, and the application of coherence measures in quantum information science.This paper introduces a rigorous framework for quantifying coherence in quantum systems. The authors define coherence as a physical resource and identify intuitive and computable measures of coherence. They establish conditions that any valid coherence measure must satisfy, distinguishing between valid and invalid measures. The paper outlines the need for a complete theory of coherence as a resource, highlighting the importance of coherence in quantum information science and its role in biological systems. The authors define incoherent states as those diagonal in a fixed basis and incoherent operations as quantum operations that map incoherent states to incoherent states. They distinguish between two types of incoherent operations: those that do not retain measurement outcomes and those that do. A maximally coherent state is defined as a state that can be used to deterministically generate all other d-dimensional quantum states through incoherent operations. The paper presents two key coherence measures: the relative entropy of coherence and the $ l_1 $-norm of coherence. These measures satisfy the required properties of monotonicity under incoherent operations and convexity. The relative entropy of coherence is defined as the difference between the von Neumann entropy of a state and the entropy of its diagonal counterpart. The $ l_1 $-norm of coherence is the sum of the absolute values of the off-diagonal elements of a density matrix. The authors also discuss other potential coherence measures, such as those based on the $ l_2 $-norm and fidelity, but show that some of these do not satisfy the required monotonicity conditions. The paper concludes with an outlook on future research directions, including the development of a complete theory of coherence as a resource, the exploration of infinite-dimensional systems, and the application of coherence measures in quantum information science.