We address the question of the exponential stability for the C-1 norm of general one-dimensional quasilinear systems with source terms under boundary conditions. To reach this aim, we introduce the notion of basic C-1- Lyapunov functions, a generic kind of exponentially decreasing function whose existence ensures the exponential stability of the system for the C-1 norm. We show that the existence of a basic C(1 )Lyapunov function is subject to two conditions: an interior condition, intrinsic to the system, and a condition on the boundary controls. We give explicit sufficient interior and boundary conditions such that the system is exponentially stable for the C-1 norm and we show that the interior condition is also necessary to the existence of a basic C-1 Lyapunov function. Finally, we show that the results conducted in this article are also true under the same conditions for the exponential stability in the C-p norm, for any p is an element of N*.