In this thesis, at first we represent some definitions and theorems which will be used in the sequel. Then, we describe the complex moving least-squares (CMLS) approximation which is constructed. In fact the on the basis of the moving least-squares (MLS) approximation in the regime of the complex variables CMLS approximation is an approximation of a vector function. After that the improved complex moving least-squares (ICMLS) approximation which is another version of the CMLS is expressed and compared with the CMLS approximation. It seems that the ICMLS version is somewhat more efficient. Then, the novel method of complex element-free Galerkin (CEFG) of Li et al. is implemented on the one- and two-dimensional nonlinear Schr?dinger equations as well as the coupled nonlinear Schr?dinger equations. The CEFG method is based on the CMLS approximation and the well-known element-free Galerkin (EFG) method. The nonlinear Schr?dinger equation is one of the famous complex nonlinear evolution equations with various and important applications in hydrodynamics, plasma physics, nonlinear optics, selffocusing in laser pulses, propagation of heat pulses in crystals, description of the dynamics of Bose–Einstein condensate at extremely low temperature and so on. In our numerical simulations, we have employed the Galerkin weak form to obtain the weak equations, while the penalty method is used to apply the essential boundary conditions. We have applied both methods of EFG and CEFG for simulating motion of single soliton solutions, interaction of two and three solitary waves and generation of solitons with the Maxwellian initial condition. Some conserved quantities are computed numerically for testing validity of the methods. It seems that the CEFG method is somewhat more efficient than the EFG method for solving considered nonlinear Schr?dinger equations. In fact the most important consideration is that for sloving Schr?dinger equations with the EFG method we should convert every obtained complex equation to two real equations, then the EFG method is implemented on these equations, while the CEFG method is directly implemented on the complex equation. So in the EFG method we should solved a system of order 2N ? 2N in every iteration but in the CEFG method we should solved a system of order N ? N .