Nine Chapters by nine eminent contributors in three parts of three chapters each, cover the fundamentals of Sobolev spaces and their applications to many areas of differential equations. The first part gives preliminary results on the LP spaces, the Fourier transforms and the theory of distributions. The second part deals with some basic results on Sobolev spaces, density theorems, approximation theorems and embedding theorems. The last part highlights some of the applications of the Sobolev spaces to the regularity theory of elliptic equations, critical point theory and error estimates in the finite element methods.