A detailed look at solving advection equations by using a Characteristic Map. Three items in this note: (1) Using the CM as the tool for advection, substeps can be generated by efficiently, in fact logarithmically fast; (2) The exact solution for the CM is given, leading to an accurate numerical algorithm for using it. Because the solution involves exponentiated 3X3 matrices, the algorithm is relatively slow although very accurate. In fact, for rigid rotations the algorithm is exact. (3) Several standard tests of advection accuracy are evaluated and the error in several popular advection schemes are quantified by comparing them with the exact solution.