Numerical methods for the solution of the Orr-Sommerfeld equations are considered. The problem is stated in §1, and the basic numerical tools (the setting up of the finite difference approximations, and the algorithm for solving the resulting algebraic eigenvalue problem) are developed in §§2 and 3. In §4 it is shown that the form in which the results are to be presented makes the actual computation into an inner-outer type iteration scheme, and that the appropriate balancing of these iterations leads to a considerable saving in computer time. Some questions bearing on the stability of the calculations are considered in §5.