The paper discusses the application of flow-based sampling to lattice field theories, addressing critical issues such as critical slowing down and topological freezing. Flow-based samplers, which use normalizing flows parameterized by machine learning models, offer promising solutions to these problems. The review covers the following key areas: 1. **Introduction**: Lattice simulations are crucial for understanding quantum field theories, but traditional Monte Carlo methods face challenges like critical slowing down and topological freezing as the continuum limit is approached. 2. **Flow-based Sampling**: - **Normalizing Flows**: These are invertible transformations between distributions, often used to efficiently sample complex distributions. - **Flows Using Machine Learning**: Techniques like discrete and continuous learnable flows are explored, with neural networks being particularly effective for parameterizing these transformations. - **Application to Scalar Field Theory**: Early demonstrations showed that flow-based models could eliminate critical slowing down in interacting scalar field theories. - **Symmetries and Gauge Theory**: Incorporating gauge symmetries into flow-based models improves training efficiency and model quality. - **Fermions and QCD**: Flow-based models have been applied to theories with fermions and in higher spacetime dimensions, including lattice QCD. 3. **Current Research**: - **New Paradigms**: Flow-based models can be used for tasks beyond direct sampling, such as estimating partition functions, parameter dependence, and solving sign problems. - **Practical Gains**: Parallel sampling and storage-free ensembles are significant benefits of flow-based methods. - **Towards QCD at Scale**: Progress has been made in applying flow-based sampling to lattice QCD, including dynamical fermions and higher-dimensional theories. 4. **Outlook**: - Future research will focus on developing more efficient and expressive flow architectures, optimizing training methods, and creating comprehensive software libraries. - Potential avenues for maximum impact include hybrid methods that combine normalizing flows with traditional Markov Chain updates, leveraging the hierarchy of scales in lattice field theory problems. The paper highlights the rapid progress in flow-based sampling and its potential to revolutionize lattice field theory calculations, particularly in addressing the challenges of critical slowing down and topological freezing.The paper discusses the application of flow-based sampling to lattice field theories, addressing critical issues such as critical slowing down and topological freezing. Flow-based samplers, which use normalizing flows parameterized by machine learning models, offer promising solutions to these problems. The review covers the following key areas: 1. **Introduction**: Lattice simulations are crucial for understanding quantum field theories, but traditional Monte Carlo methods face challenges like critical slowing down and topological freezing as the continuum limit is approached. 2. **Flow-based Sampling**: - **Normalizing Flows**: These are invertible transformations between distributions, often used to efficiently sample complex distributions. - **Flows Using Machine Learning**: Techniques like discrete and continuous learnable flows are explored, with neural networks being particularly effective for parameterizing these transformations. - **Application to Scalar Field Theory**: Early demonstrations showed that flow-based models could eliminate critical slowing down in interacting scalar field theories. - **Symmetries and Gauge Theory**: Incorporating gauge symmetries into flow-based models improves training efficiency and model quality. - **Fermions and QCD**: Flow-based models have been applied to theories with fermions and in higher spacetime dimensions, including lattice QCD. 3. **Current Research**: - **New Paradigms**: Flow-based models can be used for tasks beyond direct sampling, such as estimating partition functions, parameter dependence, and solving sign problems. - **Practical Gains**: Parallel sampling and storage-free ensembles are significant benefits of flow-based methods. - **Towards QCD at Scale**: Progress has been made in applying flow-based sampling to lattice QCD, including dynamical fermions and higher-dimensional theories. 4. **Outlook**: - Future research will focus on developing more efficient and expressive flow architectures, optimizing training methods, and creating comprehensive software libraries. - Potential avenues for maximum impact include hybrid methods that combine normalizing flows with traditional Markov Chain updates, leveraging the hierarchy of scales in lattice field theory problems. The paper highlights the rapid progress in flow-based sampling and its potential to revolutionize lattice field theory calculations, particularly in addressing the challenges of critical slowing down and topological freezing.