The numerical solution of ordinary differential equations will be the main topics discussed in the first half of this thesis. In Chapter 2 initial value problems are examined and the resulting differential equations are solved by using extrapolation techniques. Computer trials of the algorithm are completed by a special ordinary differential equation tester program and the statistics of its performance are compared with other established methods. In the following Chapter the relevant theory and properties, associated with Chebyshev polynomials, is presented. Then boundary value problems are solved by representing the ordinary differential equations as a Chebyshev series. Finally, similar methods are developed for the solution of partial differential equations.