This chapter serves as the “practice chapter” for the main goal of Part I: solving the MHD Riemann problem. Lagrangian and Eulerian frames of reference are introduced from which the three Riemann invariants of HD are identified. Space-time diagrams are introduced as a useful visual and conceptual aid in understanding the role of characteristic paths through a continuum, which is in keeping with the text’s underlying approach of treating fluid dynamics as a form of wave mechanics. The Riemann problem for HD is defined and a method of characteristics is introduced whose main purpose is to understand qualitatively how the solution to the HD Riemann problem begins to unfold. In so doing, shocks and contact discontinuities are rediscovered and rarefaction fans are introduced. It is shown how examining the eigenkets leads to profiles of the primitive variables across a rarefaction fan which ultimately leads to a semi-analytic solution to the HD Riemann problem.