The paper introduces the Generalized Approximate Message Passing (GAMP) algorithm for estimating an i.i.d. random vector observed through a linear transform followed by a probabilistic measurement channel. GAMP extends earlier approximate message passing methods to handle arbitrary input and output distributions, making it applicable to nonlinear compressed sensing and learning problems. The algorithm decouples the vector-valued estimation problem into a sequence of scalar problems and linear transforms, enabling efficient computation. The paper extends the analysis of Bayati and Montanari to show that the asymptotic behavior of GAMP under large, i.i.d. Gaussian transforms is described by a simple set of state evolution (SE) equations. These equations allow exact prediction of performance metrics like mean-squared error. The analysis applies to arbitrary input and output distributions, even for non-convex optimization problems. GAMP matches predictions from relaxed belief propagation and agrees with optimal performance from the replica method in statistical physics. The GAMP algorithm is a generalization of approximate message passing methods, providing efficient approximations of both max-sum and sum-product loopy belief propagation. It is computationally simple, with each iteration involving scalar estimation and linear transforms. The algorithm's performance is validated through simulations, showing excellent convergence in a small number of iterations. The paper discusses various applications of GAMP, including AWGN output channels, non-Gaussian noise models, logistic channels, and discrete distributions. It also presents scalar estimation functions for approximating loopy BP, enabling MAP and MMSE estimation. The algorithm is shown to reduce vector-valued problems to scalar problems, with effective Gaussian noise levels represented by scalar parameters. The asymptotic analysis of GAMP for large, Gaussian i.i.d. matrices A is presented, showing that the algorithm's behavior is described by SE equations. These equations allow precise asymptotic performance guarantees and are validated through simulations. The paper concludes that GAMP provides a computationally efficient and widely applicable methodology for estimation with linear mixing, with exact asymptotic performance characterizations for large random matrices.The paper introduces the Generalized Approximate Message Passing (GAMP) algorithm for estimating an i.i.d. random vector observed through a linear transform followed by a probabilistic measurement channel. GAMP extends earlier approximate message passing methods to handle arbitrary input and output distributions, making it applicable to nonlinear compressed sensing and learning problems. The algorithm decouples the vector-valued estimation problem into a sequence of scalar problems and linear transforms, enabling efficient computation. The paper extends the analysis of Bayati and Montanari to show that the asymptotic behavior of GAMP under large, i.i.d. Gaussian transforms is described by a simple set of state evolution (SE) equations. These equations allow exact prediction of performance metrics like mean-squared error. The analysis applies to arbitrary input and output distributions, even for non-convex optimization problems. GAMP matches predictions from relaxed belief propagation and agrees with optimal performance from the replica method in statistical physics. The GAMP algorithm is a generalization of approximate message passing methods, providing efficient approximations of both max-sum and sum-product loopy belief propagation. It is computationally simple, with each iteration involving scalar estimation and linear transforms. The algorithm's performance is validated through simulations, showing excellent convergence in a small number of iterations. The paper discusses various applications of GAMP, including AWGN output channels, non-Gaussian noise models, logistic channels, and discrete distributions. It also presents scalar estimation functions for approximating loopy BP, enabling MAP and MMSE estimation. The algorithm is shown to reduce vector-valued problems to scalar problems, with effective Gaussian noise levels represented by scalar parameters. The asymptotic analysis of GAMP for large, Gaussian i.i.d. matrices A is presented, showing that the algorithm's behavior is described by SE equations. These equations allow precise asymptotic performance guarantees and are validated through simulations. The paper concludes that GAMP provides a computationally efficient and widely applicable methodology for estimation with linear mixing, with exact asymptotic performance characterizations for large random matrices.