The aim here is to discuss the existence and long time behavior of BV solutions to the Cauchy problem for (possibly inhomogeneous) strictly hyperbolic systems of balance laws. Thus, this chapter may be viewed as the counterpart of Section 5.5, where the same issues are addressed in the context of classical solutions. For the reasons presented in the preceding chapters, the investigation shall be confined to systems in a single spatial dimension and initial data of small total variation; however, modulo these limitations, the analogy to the results of Section 5.5 goes quite far.