This thesis deals with the numerical solution of the transient heat conduction problem by using the finite element method and the improved Euler method. In the space domain, the finite element method is used. In the time domain of integration, the improved Euler method is employed. Two examples are demonstrated in this thesis. The first example is a test problem designed to verify the correctness of the computer code. One finds that the combination of the 6 point triangular element or 9 point quadrilateral element with the improved Euler method may be used to obtain both accurate and stable results. The second example is a practical engineering problem--to find the steady state and transient temperature distributions in a turbine blade. The computation results show that the transient solution converges to the steady state solution after a proper period of time. In addition, a general finite element code is developed in the thesis to solve the time-dependent problems. (Abstract shortened with permission of author.)