The paper addresses the limitations of differential privacy, particularly in terms of utility guarantees and the inability to handle complex machine learning tasks. It introduces a new approach called metric privacy, which generalizes differential privacy and allows for more flexible and accurate synthetic data generation. The authors develop a polynomial-time algorithm to create a private measure from a dataset, which can then be used to generate synthetic data that preserves statistical properties while maintaining privacy. They prove an asymptotically sharp min-max result for private measures and synthetic data in compact metric spaces, showing that the accuracy of the synthetic data is optimal. A key component of their construction is a new superregular random walk, which has a joint distribution as regular as independent random variables but deviates from the origin logarithmically slowly. This random walk is used to construct a private measure on the interval and then extended to general metric spaces, demonstrating the effectiveness of their method in various applications.The paper addresses the limitations of differential privacy, particularly in terms of utility guarantees and the inability to handle complex machine learning tasks. It introduces a new approach called metric privacy, which generalizes differential privacy and allows for more flexible and accurate synthetic data generation. The authors develop a polynomial-time algorithm to create a private measure from a dataset, which can then be used to generate synthetic data that preserves statistical properties while maintaining privacy. They prove an asymptotically sharp min-max result for private measures and synthetic data in compact metric spaces, showing that the accuracy of the synthetic data is optimal. A key component of their construction is a new superregular random walk, which has a joint distribution as regular as independent random variables but deviates from the origin logarithmically slowly. This random walk is used to construct a private measure on the interval and then extended to general metric spaces, demonstrating the effectiveness of their method in various applications.