The paper discusses the use of the L-curve in regularization methods for solving discrete ill-posed problems. The L-curve is a plot of the size of the regularized solution versus the size of the residual for all valid regularization parameters. Two main results are established: a unifying characterization of various regularization methods, where the measurement of "size" depends on the chosen method (e.g., 2-norm for Tikhonov regularization, 1-norm in the SVD coordinate system for truncated SVD regularization); and a new method for choosing the regularization parameter based on the L-curve, which is shown to be more robust in the presence of correlated errors compared to generalized cross-validation. The paper also investigates the properties of the L-curve, including its shape and how it can be used to distinguish signal from noise. Numerical examples illustrate the effectiveness of the L-curve criterion in choosing the regularization parameter, demonstrating its robustness and reliability in various scenarios.The paper discusses the use of the L-curve in regularization methods for solving discrete ill-posed problems. The L-curve is a plot of the size of the regularized solution versus the size of the residual for all valid regularization parameters. Two main results are established: a unifying characterization of various regularization methods, where the measurement of "size" depends on the chosen method (e.g., 2-norm for Tikhonov regularization, 1-norm in the SVD coordinate system for truncated SVD regularization); and a new method for choosing the regularization parameter based on the L-curve, which is shown to be more robust in the presence of correlated errors compared to generalized cross-validation. The paper also investigates the properties of the L-curve, including its shape and how it can be used to distinguish signal from noise. Numerical examples illustrate the effectiveness of the L-curve criterion in choosing the regularization parameter, demonstrating its robustness and reliability in various scenarios.