The distinguishing number of a graph was introduced by Albertson and Collins as a measure of the amount of symmetry contained in the graph. Tymoczko extended this definition to faithful group actions on sets; taking the set to be the vertex set of a graph and the group to be the automorphism group of the graph allows one to recover the previous definition. Since then, several authors have studied properties of the distinguishing number as well as extensions of the notion. In this paper, we first answer a few open questions regarding the distinguishing number. Next we turn to generalizations regarding the labeling of Cartesian powers of a set and the different subgroups that can be obtained through labelings. We then introduce a new partially ordered set on partitions that follows naturally from extending the theory of distinguishing numbers to that of distinguishing partitions. Then we investigate the groups obtainable from partitioning Cartesian powers of a set in more detail and show how the original notion of the distinguishing number of a graph can be recovered in this way. Next, we introduce a polynomial and a symmetric function generalization of the distinguishing number. Finally, we present a large number of open questions and problems for further research.