Abstarct: This dissertation deals with the numerical analysis of viscoelastic and elastic structures. In particular, Finite Element Method (FEM) has been utilized in the analysis of viscoelastic structures, and the analysis of elastic structures have been performed by Boundary Element Method (BEM). The main motivation behind this research has been to enable accurate analysis of structures subjected to mechanical loading with the reduction in computational cost compared with conventional methods. In order to achieve this goal, it is proposed that the novel elements have been substituted for classical elements in the analysis. In particular, complex Fourier elements have been employed in the FE analysis of viscoelastic structures. Moreover, Extended Multiquadric (XMQ) elements and Gaussian-Fourier elements have been introduced and utilized in the FE analysis of 2D viscoelastic problems. In addition, FE analysis of thin viscoelastic plates has been performed by proposing new plate elements which are called Discrete Kirchhoff- Fourier Triangular (DKFT) plate elements. And finally, XMQ elements have been used in the BE analysis of Elastostatics problmes. Point interpolation method baxsed on radial basis functions (RPIM) has been employed in order to obtain shape functions in these new elements.