Measure‐valued weak solutions for conservation laws with discontinuous flux are proposed and explicit formulae have been derived. We propose convergent discontinuous flux‐based numerical schemes for the class of hyperbolic systems that admit nonclassical urn:x-wiley:00222526:media:sapm12319:sapm12319-math-0001‐shocks, by extending the theory of discontinuous flux for nonlinear conservation laws to scalar transport equation with a discontinuous coefficient. The article also discusses the concentration phenomenon of solutions along the line of discontinuity, for scalar transport equations with a discontinuous coefficient. The existence of the solutions for transport equation is shown using the vanishing viscosity approach and the asymptotic behavior of the solutions is also established. The performance of the numerical schemes for both scalar conservation laws and systems to capture the urn:x-wiley:00222526:media:sapm12319:sapm12319-math-0002‐shocks effectively is displayed through various numerical experiments.