Metamaterials (MTMs), which are formed by embedding inclusions and material components in host media to achieve composite media that may be engineered to have qualitatively new physically realizable response functions that do not occur or may not be easily available in nature, have raised a great deal of interest in recent years. In this paper, we highlight a large variety of the physical effects associated with double- and single-negative MTMs and some of their very interesting potential applications. The potential ability to engineer materials with desired electric and magnetic properties to achieve unusual physical effects offers a great deal of excitement and promise to the scientific and engineering community. While some of the applications we will discuss have already come to fruition, there are many more yet to be explored.