It is well known that the Cauchy problem for the system (1.1.1), satisfying the initial condition u(x, 0) = uo(x),x ∈ IRm does not have, in general, a smooth solution beyond some finite time interval, even when u0 is sufficiently smooth. For this reason, we study the weak solutions containing the discontinuities; these weak solutions are, in general, not unique, and one needs an admissibility criterion to select physically relevant solutions. Here, we study the one-dimensional scalar case to highlight some of these fundamental issues in the theory of hyperbolic equations of conservation laws.