In this thesis, we extend the spectral collocation methods(SCM) (i.e., pseudo-spectral method) in Quarteroni and Valli [27] for the semilinear, parameter-dependentproblems(PDP) in the square with the Dirichlet boundary condition. The optimal error bounds are derived in this thesis for both H1 and L2 norms. For the solutions sufficiently smooth, the very high convergence rates can be obtained. The algorithms of the SCM are simple and easy to carry out. Only a few of basis functions are needed so that not only can the high accuracy of the PDP solutions be achieved, but also a great deal of CPU time may be saved. Moreover, for PDP the stability analysis of SCM is also made, to have the same growth rates of condition number as those for Poisson¡¦s equation. Numerical experiments are carried out to verify the theoretical analysis made.