Error estimation in finite element method has been the subject of the research of many scientists over the recent years. In order to improve finite element ability, error estimation and adaptivity are employed to optimize mesh configuration and shape functions. In the present thesis, the adaptivity method is used to achieve clear optimum topology for plates. The main steps in the present thesis involve the following: a) Thickness optimization of plates in order to obtain maximum natural frequencies and critical loads. b) Error estimation of eigenproblem solution in order to refine the mesh and decreasing error. c) Obtaining the optimum and clear topology of plates using optimization and adaptivity technique. To achieve above purposes, at first, vibration and stability analysis of plates are presented and then the first order optimization technique is used to optimize plates topology. For this, plate thickness and mass constancy are considered as a design variable and main limit, respectively. A procedure is proposed to estimate the error of eigenproblem solution. Based on the calculated error and slope of optimized thickness, the mesh is refined. Finally, both optimization and adaptivity techniques are used to illustrate the scope and efficiency of the procedure by solving several examples.