On the Approximation of Conservation Laws by Vanishing Viscosity

In this thesis I collect some recent results on the approximation of conservation laws by vanishing viscosity. The exposition is organized as follows. In the first chapter I provide a general introduction to the motivations of my work and a short overview of the results I obtained, I recall the tools and the main ideas involved in the proves, and I give the main references. Chapters 2 and 3 are devoted to present the results I obtained in collaboration with prof. A. Bressan, dealing with the viscous approximation of a scalar conservation law, [10], [11].