The paper introduces a novel algorithm called Generalized Approximate Message Passing (GAMP) for estimating random vectors observed through a linear transform followed by a component-wise, probabilistic measurement channel. GAMP provides computationally efficient approximate implementations of max-sum and sum-product belief propagation for such problems. The algorithm extends earlier approximate message passing methods to handle arbitrary distributions on both the input and output of the transform, making it applicable to a wide range of problems in nonlinear compressed sensing and learning. The paper argues that for large, i.i.d. Gaussian transforms, the asymptotic behavior of GAMP can be described by a set of state evolution (SE) equations. From these equations, one can predict the asymptotic value of various performance metrics, such as mean-squared error or detection accuracy. The analysis is valid for arbitrary input and output distributions, even when the optimization problems are non-convex. The results match predictions from relaxed belief propagation on large sparse matrices and, in some cases, agree with optimal performance predicted by the replica method in statistical physics. GAMP provides a general and systematic approach to a large class of estimation problems with linear mixing, offering computational tractability and precise asymptotic performance guarantees. The paper includes examples of linear mixing problems, introduces the GAMP algorithm, and presents detailed analyses of its performance and asymptotic behavior.The paper introduces a novel algorithm called Generalized Approximate Message Passing (GAMP) for estimating random vectors observed through a linear transform followed by a component-wise, probabilistic measurement channel. GAMP provides computationally efficient approximate implementations of max-sum and sum-product belief propagation for such problems. The algorithm extends earlier approximate message passing methods to handle arbitrary distributions on both the input and output of the transform, making it applicable to a wide range of problems in nonlinear compressed sensing and learning. The paper argues that for large, i.i.d. Gaussian transforms, the asymptotic behavior of GAMP can be described by a set of state evolution (SE) equations. From these equations, one can predict the asymptotic value of various performance metrics, such as mean-squared error or detection accuracy. The analysis is valid for arbitrary input and output distributions, even when the optimization problems are non-convex. The results match predictions from relaxed belief propagation on large sparse matrices and, in some cases, agree with optimal performance predicted by the replica method in statistical physics. GAMP provides a general and systematic approach to a large class of estimation problems with linear mixing, offering computational tractability and precise asymptotic performance guarantees. The paper includes examples of linear mixing problems, introduces the GAMP algorithm, and presents detailed analyses of its performance and asymptotic behavior.