After an introductory mathematical review of the general concept of self-similarity with respect to a given rescaling algebra, attention is focused on the case of Newtonian systems in Galilean space-time, whose self-similarity transformations will form subgroups of the maximal self-similarity group of the Galilean space-time structure itself. As a prerequisite for a systematic general investigation of self-similarity in such Newtonian systems, it is shown how an appropriately adapted similarity transporting coordinate system can be constructed explicitly for an arbitrary generator of the 12 parameter Galilean self-similarity group. A concrete application is provided by the case of the Keplerian disk, which is kinematically self-similar under the action of a three parameter subgroup. It is shown in the explicit example of a Eulerian fluid system how the generic self-similarity problem can be formulated as an effective stationarity problem.