This book, a revision of Radford M. Neal's Ph.D. thesis, explores the Bayesian approach to learning flexible statistical models based on neural networks. The author aims to provide theoretical insights and practical utility, addressing the challenges of complexity and computational demands in Bayesian learning for neural networks. Chapter 1 introduces the Bayesian framework, neural network models, and Markov chain Monte Carlo methods. Chapter 2 defines priors for network parameters that converge to Gaussian processes or non-Gaussian stable processes, showing that Bayesian learning does not require limiting model complexity. Chapter 3 discusses the computational challenges using Markov chain Monte Carlo methods, particularly the hybrid Monte Carlo algorithm. Chapter 4 evaluates Bayesian neural network models on synthetic and real data sets, demonstrating their effectiveness in handling large networks and automatic relevance determination. Chapter 5 concludes with a discussion of related work and future directions. The book also includes software for implementing the methods, though it is not intended for routine data analysis.This book, a revision of Radford M. Neal's Ph.D. thesis, explores the Bayesian approach to learning flexible statistical models based on neural networks. The author aims to provide theoretical insights and practical utility, addressing the challenges of complexity and computational demands in Bayesian learning for neural networks. Chapter 1 introduces the Bayesian framework, neural network models, and Markov chain Monte Carlo methods. Chapter 2 defines priors for network parameters that converge to Gaussian processes or non-Gaussian stable processes, showing that Bayesian learning does not require limiting model complexity. Chapter 3 discusses the computational challenges using Markov chain Monte Carlo methods, particularly the hybrid Monte Carlo algorithm. Chapter 4 evaluates Bayesian neural network models on synthetic and real data sets, demonstrating their effectiveness in handling large networks and automatic relevance determination. Chapter 5 concludes with a discussion of related work and future directions. The book also includes software for implementing the methods, though it is not intended for routine data analysis.