Our present understanding of how the interactions between electrons affect the metallic state has, for forty years, rested on the foundations of Landau's Fermi-liquid theory. It provides the basis for understanding metals in terms of weakly interacting electron (-like) particles. Recent years have seen the discovery of metals which appear to fall outside this framework-perhaps most notably in the normal state of the high temperature cuprate superconductors. While the theory for understanding the cuprate metals remains controversial, there are a number of clear examples where we do believe we understand the new underlying theoretical concepts. In this article I illustrate four such routes towards forming a non-Fermi liquid metal and illustrate, where possible, how these have been realized in a number of materials. The proximity to a quantum phase transition and reduced effective dimensionality can both play important roles.