A literature review regarding the use of the R-functions theory to solve linear and nonlinear dynamics problems of the functionally graded plates and shallow shells is presented in the work. The paper reviews the main works devoted to analysis of the functionally graded structures with an arbitrary planform and different boundary conditions. In most cases the R-functions theory is applied together with the Ritz method. Therefore questions connected with construction of sets of admissible functions that have to satisfy at least the geometric (essential) boundary conditions are discussed also here. The review focuses on papers using the classical and first order shear deformation theories applied widely in the modeling of laminated and functionally graded plates and shells. In order to demonstrate the effectiveness and universalities of the R-functions method (RFM), some numerical results related to dynamic behavior and stability of the functionally graded plates and shallow shells are presented.