In this paper we present the concept of a matrix stability preserving map and show its impact on the problem of robust controller design. We develop a number of tests for checking whether a given matrix is a stability preserving map. We show that the concept of a stability preserving map can be used to provide a different characterization of the existence of a fixed order controller that simultaneously stabilizes a finite number of plants. We also demonstrate how it can be used to state conditions for the robust stabilization of families of plants with real parameter uncertainty. In addition, we show how stability preserving map tests lead to robust stabilization techniques and apply the methodology to a number of examples.