This paper presents a sharp statistical theory for conditional diffusion models (CDMs) with classifier-free guidance. CDMs are widely used in image synthesis, reinforcement learning, and other fields, where they incorporate conditional information to guide sample generation. Despite their empirical success, the theoretical understanding of CDMs is limited. This work fills this gap by establishing a statistical theory for distribution estimation using CDMs. The theory provides a sample complexity bound that adapts to the smoothness of the data distribution and matches the minimax lower bound. The key to the theoretical development is a novel approximation result for the conditional score function, which relies on a diffused Taylor approximation technique. The statistical theory is applied to elucidate the performance of CDMs in various applications, including model-based transition kernel estimation in reinforcement learning, solving inverse problems, and reward-conditioned sample generation. The paper establishes the first universal approximation theory of conditional score functions using neural networks. It shows that the network size scales adaptively to the smoothness of the data distribution. The results are built upon a novel conditional score approximation theory, which develops a diffused Taylor approximation technique. The paper also provides sample complexity bounds for distribution estimation using CDMs. The analysis is based on a bias-variance trade-off in nonparametric statistics and connects to the sample complexity bound via Girsanov's theorem from stochastic processes. The statistical rate matches the minimax lower bound. The paper also provides statistical guarantees for applying CDMs to model-based reinforcement learning. The paper further establishes theoretical foundations of CDMs for solving inverse problems and reward-conditioned sample generation. It demonstrates the utility of the established statistical theories. Specifically, it presents sub-optimality bounds when generating high-reward samples in an offline setting and error bounds for estimating the posterior mean given a measurement in linear inverse problems. These results theoretically explain the performance of CDMs. The paper contributes to the theory of diffusion models and develops the first set of theories of CDMs trained with classifier-free guidance.This paper presents a sharp statistical theory for conditional diffusion models (CDMs) with classifier-free guidance. CDMs are widely used in image synthesis, reinforcement learning, and other fields, where they incorporate conditional information to guide sample generation. Despite their empirical success, the theoretical understanding of CDMs is limited. This work fills this gap by establishing a statistical theory for distribution estimation using CDMs. The theory provides a sample complexity bound that adapts to the smoothness of the data distribution and matches the minimax lower bound. The key to the theoretical development is a novel approximation result for the conditional score function, which relies on a diffused Taylor approximation technique. The statistical theory is applied to elucidate the performance of CDMs in various applications, including model-based transition kernel estimation in reinforcement learning, solving inverse problems, and reward-conditioned sample generation. The paper establishes the first universal approximation theory of conditional score functions using neural networks. It shows that the network size scales adaptively to the smoothness of the data distribution. The results are built upon a novel conditional score approximation theory, which develops a diffused Taylor approximation technique. The paper also provides sample complexity bounds for distribution estimation using CDMs. The analysis is based on a bias-variance trade-off in nonparametric statistics and connects to the sample complexity bound via Girsanov's theorem from stochastic processes. The statistical rate matches the minimax lower bound. The paper also provides statistical guarantees for applying CDMs to model-based reinforcement learning. The paper further establishes theoretical foundations of CDMs for solving inverse problems and reward-conditioned sample generation. It demonstrates the utility of the established statistical theories. Specifically, it presents sub-optimality bounds when generating high-reward samples in an offline setting and error bounds for estimating the posterior mean given a measurement in linear inverse problems. These results theoretically explain the performance of CDMs. The paper contributes to the theory of diffusion models and develops the first set of theories of CDMs trained with classifier-free guidance.