We shall consider some approximation methods in the finite dimensional setting for obtaining stable approximate solutions for ill-posed operator equations in the infinite dimensional setting. The methods are essentially devised by replacing some or all of the operators appearing in the Tikhonov-regularized equation by certain finite rank operators. Results on convergence and error estimates are derived by choosing the regularization parameter and approximation parameter appropriately. The discussed methods are applied to Fredholm integral equations of the first kind which is an ill-posed problem. These results are mainly based on some of the publications of the author and his collaborators.