The Method of Transport (MoT) developed by Fey (JCP 1998) and modified by Noelle (JCP 2000) is a Riemann-solver-free scheme for multidimensional systems of hyperbolic conservation laws arising in fluid mechanics. Zimmermann and Noelle have shown that for the Euler equations,the MoT may be derived from a kinetic-type decomposition of the system into single advection equations which are coupled via a right-hand side, which encodes the interaction between the waves. For each of these advection equations, one can derive an exact evolution operator, and the MoT is an approximation to these evolution operators. While the numerical treatment of the advction operator is by now well-understood, the discretisation of the interaction term still raises some challenging questions. Recent numerical experiments of the authors show that current versions of the MoT may introduce unphysical acoustic waves in the presence of linear discontinuities. This seems to indicate that the MoT in its current state is not suitable for the computation of at least some of the discontinuities which arise naturally in fluid mechanics. Note that the derivation of the evolution operators assumes that the solution is smooth. Hence it might be possible to find more general evolution operators valid for weak solutions as well. This might lead to new versions of the MoT more suitable for the computation of discontinous solutions. All of this is work in progress.