This talk will provide an introduction into a class of infinite permutation groups which has been introduced by myself in my PhD thesis, for an english translation see here. Namely - as many of you will guess - the talk is about the class of residue class-wise affine groups. In contrast to infinite permutation groups in general, explicit machine computation in such groups is quite feasible -- see my GAP package RCWA. I will discuss the key results of my thesis and give a (very brief!) outline of how I have obtained them. In the sequel I will give a few examples. The talk will end with a brief outlook on ideas concerning possible further work on the subject. The latter will comprise a way to `turn the Collatz tree into a complete infinite binary tree', representations of Grigorchuk's groups and - if time permits - considerations motivated by Sarkovskii's Theorem. The talk will not provide information about the algorithmic aspects of the topic. In particular, nothing will be said about the algorithms implemented in RCWA.