This paper provides a comprehensive review of normalizing flows, a powerful tool for constructing flexible probability distributions. Normalizing flows are defined by a series of invertible and differentiable transformations applied to a simple base distribution, allowing for the creation of complex, multi-modal distributions. The review covers the fundamental principles of flow design, including expressive power and computational trade-offs, and discusses various methods for constructing flows, such as autoregressive flows, combination-based transformers, integration-based transformers, and spline-based transformers. The paper also explores the application of normalizing flows in tasks like generative modeling, approximate inference, and supervised learning, highlighting their versatility and potential for future research.This paper provides a comprehensive review of normalizing flows, a powerful tool for constructing flexible probability distributions. Normalizing flows are defined by a series of invertible and differentiable transformations applied to a simple base distribution, allowing for the creation of complex, multi-modal distributions. The review covers the fundamental principles of flow design, including expressive power and computational trade-offs, and discusses various methods for constructing flows, such as autoregressive flows, combination-based transformers, integration-based transformers, and spline-based transformers. The paper also explores the application of normalizing flows in tasks like generative modeling, approximate inference, and supervised learning, highlighting their versatility and potential for future research.