Ultrasonic guided waves are frequently employed for Non-destructive tests (NDT), and Structural Health Monitoring (SHM) applications in the industry. Several analytical and numerical approaches have been developed in order to investigate guided wave behavior on multilayered structures. In this thesis, guided waves were investigated using the Semi-Analytical Finite Element (SAFE) approach on planar (plate like) structures. The guided wave theory and dispersive behavior in bounded structure were presented for Lamb waves for isotropic elastic plate model first. The numerical method to solve the SAFE problem was developed using the Matlab 2014b in order to obtain dispersion curves. The curves were compared with the Disperse Software, which utilizes global matrix approach for analytical solution of the same problem. Good agreement was achieved on obtained dispersion curves. As a second part of this study, dispersion analysis was carried out in a multilayer plate model consisting of elastic and viscoelastic materials. The semi-analytical finite element method was solved by adapting the hysteretic damping model so that it can also be applied in a damped plate model. Phase velocity and attenuation dispersion curves were illustrated and the effect of the viscoelastic layer thickness is also discussed. The obtained attenuation dispersion curves in this damped plate configuration are examined for wave modes with low-attenuation. The dispersion curve results obtained using the SAFE method was also compared with the results of studies available in literature.