Viscous conservation laws are the basic models for the dissipative
phenomena. We aim at a systematic presentation of the basic ideas for the quantitative study of the nonlinear waves for viscous conservation laws. The present paper concentrates on the scalar laws; an upcoming Part II will deal with the systems. The basic ideas for scalar viscous conservation laws originated from two sources: the theory for the hyperbolic conservation laws and the Burgers equation. We have initiated the Green’s function approach. These ideas are streamlined, simplified and synthesized here. We then apply them to some new problems and raise open problems. Quantitative understanding is necessary for further studies of the richer wave phenomena of the coupling of distinct wave types and the coupling of the boundary with the nonlinear waves. Viscous conservation laws may be viewed as the basic models for general dissipative systems.