Plates and shells are excellent structural elements. Analys of plates and shells is an important branch in modern solid mechanics. It plays a guiding role in many fields because of its wide application to almost all the engineering design, especially to astronautics, aeronautics, marine, machinery, petrochemical industry, architecture, water conservancy, power, instruments and transportation. Analysis of plates and shells originates from the 18th century with the development of industry. In the 20th century , the rocketing development of industry greatly stimulated the development and application of this subject. Now, the classical linear theory of thin plates and shells has matured and already been playing a decisive role in many engineering designs. However, there are still many problems left to be solved in the fields on nonlinear theory of thin plates and shells, and linear theory of thick plates and shells. Based on introduction of the history of development of this subject, this paper gives a brief account of the exploration the author did in nearly forty years, which has been well applied to engineering problems, in the areas of nonlinear bending, stability and vibration of thin plates and shells such as corrugated plates and shells, one-layer plates and shells, bimetallic shallow shells of revolution, latticed shallow shells, sandwich plates and shells, and laminated composite plates and shells. The paper also gives an introduction of the author's work on the linear bending of both thick and thin plates and shells.