In this chapter we study a technique for solving the generalized Riemann problem for systems of non–linear hyperbolic partial differential equations with source terms. The generalized Riemann problem studied here is a twofold generalization of the classical Riemann problem studied in previous chapters, namely: (i) the two vector fields that define the initial conditions are arbitrary but smooth away from the interface and (ii) the governing hyperbolic equations include source terms. We note that in the literature this Cauchy problem has also been termed Derivative Riemann Problem and High–Order Riemann Problem. In this book we shall adopt the terminology Generalized Riemann Problem and will be denoted by GRP K . Here K is an arbitrary non–negative integer and stands for the order of the approximation.