1. Background of vibration -- 2. Methods of analysis for vibration problems -- 3. Vibration basics for plates -- 4. Exact, series-type, and approximate methods for transverse vibration of plates -- 5. Development of characteristic orthogonal polynomials (COPs) in vibration problems -- 6. Boundary characteristic orthogonal polynomials (BCOPs) in vibration of plates -- 7. Transverse vibration of elliptic and circular plates -- 8. Triangular plates -- 9. Rectangular and skew plates -- 10. Circular annular and elliptic annular plates -- 11. Plates with nonhomogeneous material properties -- 12. Plates with variable thickness -- 13. Plates with orthotropic material properties -- 14. Plates with hybrid complicating effects.