This paper provides a comprehensive survey of permutation-invariant neural networks and their applications in approximating set functions. The authors highlight the importance of permutation invariance in set-based data processing, where the order of elements in a set does not affect the output. They discuss various neural network architectures, including Deep Sets, PointNet, Set Transformer, and others, which are designed to handle set-based inputs. The paper also explores the theoretical foundations of these models, including sum-decomposability and Janossy pooling, and their implications for model generalization and performance. The authors emphasize that permutation-invariant models are crucial for tasks involving sets, such as point cloud processing, set retrieval, set generation, and set matching. They also discuss the challenges and limitations of these models, as well as their potential applications in various domains. The paper concludes with a discussion on the future directions of research in permutation-invariant neural networks.This paper provides a comprehensive survey of permutation-invariant neural networks and their applications in approximating set functions. The authors highlight the importance of permutation invariance in set-based data processing, where the order of elements in a set does not affect the output. They discuss various neural network architectures, including Deep Sets, PointNet, Set Transformer, and others, which are designed to handle set-based inputs. The paper also explores the theoretical foundations of these models, including sum-decomposability and Janossy pooling, and their implications for model generalization and performance. The authors emphasize that permutation-invariant models are crucial for tasks involving sets, such as point cloud processing, set retrieval, set generation, and set matching. They also discuss the challenges and limitations of these models, as well as their potential applications in various domains. The paper concludes with a discussion on the future directions of research in permutation-invariant neural networks.