Abstract: We present some results for Radon measure-valued solutions of first order scalar conservation laws. In particular we discuss the case in which the singular part of the initial datum is a superposition of Dirac masses. In this case it is necessary to give an entropy formulation for Radon measure solutions. The main point is that the negative and the positive singular part of the measure have support along given characteristics. On the support of the singular part of the measure we have to impose, for the regular part, some compatibility conditions. When the flux is bounded the entropy formulation and the compatibility conditions are enough to characterize a unique solution. In the case of unbounded fluxes the positive and the negative singular part of the measure can intersect in a finite time and the dynamics is not clear. Therefore, we consider a proper approximation of the original problem in order to determine which are the natural conditions to be imposed. In this way it is possible to give a formulation for which the problem is well posed. The results are obtained in collaboration with M. Bertsch, F. Smarrazzo and A. Tesei.