The paper presents Multi-Level Adaptive Solutions to Boundary-Value Problems, proposing a method that combines discretization and solution processes to solve partial differential equations efficiently. The method uses a sequence of grids (levels) of increasing fineness, enabling the solution of large systems of equations in O(n) operations, where n is the number of points on the finest grid. This approach allows for adaptive discretization, where mesh sizes and approximation orders are adjusted to the solution's behavior, leading to high accuracy with minimal computational effort. The method is applicable to both linear and nonlinear problems, including elliptic and mixed-type (transonic flow) problems. The Multi-Grid (MG) method is a key component, using coarser grids to accelerate convergence by smoothing out high-frequency error components. The FAS (Full Approximation Storage) algorithm extends this to nonlinear problems, storing full approximations on each grid level. The method is efficient, with computational work predictable through local mode analysis, and is suitable for both uniform and nonuniform grids. Adaptive discretization techniques further enhance the method by dynamically adjusting mesh sizes and orders based on the solution's needs, ensuring optimal accuracy and efficiency. The paper also discusses the application of these techniques to initial-value problems and provides numerical experiments confirming the method's effectiveness. Overall, the approach significantly improves the efficiency and accuracy of solving boundary-value problems.The paper presents Multi-Level Adaptive Solutions to Boundary-Value Problems, proposing a method that combines discretization and solution processes to solve partial differential equations efficiently. The method uses a sequence of grids (levels) of increasing fineness, enabling the solution of large systems of equations in O(n) operations, where n is the number of points on the finest grid. This approach allows for adaptive discretization, where mesh sizes and approximation orders are adjusted to the solution's behavior, leading to high accuracy with minimal computational effort. The method is applicable to both linear and nonlinear problems, including elliptic and mixed-type (transonic flow) problems. The Multi-Grid (MG) method is a key component, using coarser grids to accelerate convergence by smoothing out high-frequency error components. The FAS (Full Approximation Storage) algorithm extends this to nonlinear problems, storing full approximations on each grid level. The method is efficient, with computational work predictable through local mode analysis, and is suitable for both uniform and nonuniform grids. Adaptive discretization techniques further enhance the method by dynamically adjusting mesh sizes and orders based on the solution's needs, ensuring optimal accuracy and efficiency. The paper also discusses the application of these techniques to initial-value problems and provides numerical experiments confirming the method's effectiveness. Overall, the approach significantly improves the efficiency and accuracy of solving boundary-value problems.