The Dynamic Finite Element (DFE) method is a well-established superconvergent semianalytical method that has been used in the past to investigate the vibration behaviour of various beam-structures. Considered as a viable alternative to conventional FEM for preliminary stage modal analysis, the DFE method has consistently proven that it is capable of producing highly accurate results with a very coarse mesh; a feature that is attributed to the fact that the DFE method uses trigonometric, frequency-dependant shape functions that are based on the exact solution to the governing differential equation as opposed to the polynomial shape functions used in conventional FEM. In the past many researchers have contributed towards building a comprehensive library of DFE models for various line structural elements and configurations, which would serve as the building blocks that would help the DFE method evolve into a fullfledged, versatile tool like conventional FEM in the future. However, thus far a DFE formulation has not been developed for plate problems. Therefore, in this thesis an effort has been made for the first time to develop a DFE formulation for the realm of two-dimensional structural problems by formulating a Quasi-Exact Dynamic Finite Element (QDFE) solution to investigate the free vibration behaviour of thin single- and multi-layered, rectangular plates. As a starting point for this work, Hamiltonian mechanics and the Classical Plate Theory (CPT) are used to develop the governing differential equation for thin plates. Subsequently, a unique quasiexact solution to the governing equation is sought by following a distinct procedure that, to the best of the author‘s knowledge, has never been presented before. Through this procedure, the characteristic equation is re-arranged as the sum of two beam-like expressions and then solved for by applying the quadratic formula. The resulting quasi-exact roots are then exploited to form the trigonometric basis functions, which in turn are used to derive the frequency-dependant shape functions; the characteristic feature of the QDFE method. Once developed, the new QDFE technique is applied to determine the vibration behaviour of thin, isotropic, linearly elastic, rectangular, homogenous plates. Subsequently, it is also employed to formulate a Simplified Layerwise Quasi-Exact Dynamic Finite Element solution for the free vibration of thin, rectangular multilayered plates. In addition, the quasi-exact solution to the plate equation is also utilised to develop a Dynamic Coefficient Matrix (DCM) method to investigate the vibrational characteristics of thin, rectangular, homogeneous plates and thin, rectangular, multilayered plates. The Method of Homogenization is used as an alternative procedure to validate the results from the Simplified Layerwise Quasi-Exact Dynamic Finite Element method and the Simplified Layerwise Dynamic Coefficient Matrix method. The results from both the QDFE and DCM methods are, in general, verified for accuracy against the exact results existing in the open literature and those produced by two in-house developed conventional FEM codes and/or ANSYS® software.