The authors consider the differential equation \[ {\partial u\over\partial t}+{\partial\over\partial x} \{f(u)\}= {\partial\over\partial x} \Biggl\{A \Biggl(b(u){\partial u\over\partial x}\Biggr)\Biggr\}, A(s)= \int^s_0 a(\zeta) d\zeta, a(s),b(s)\geq 0, u(x,0)= u_0(x). \] They discuss the possibility of shocks arising and the associated entropy condition. Discrete approximations involving three-point implicit difference schemes are introduced and regularity estimates are obtained. It is shown that the schemes suggested give approximate solutions which converge to a weak solution.