CMOS technology and its sustainable scaling have been the enablers for the design and manufacturing of computer architectures that have been fuelling a wider range of applications. Today, however, both the technology and the computer architectures are suffering from serious challenges/ walls making them incapable to deliver the right computing power at pre-defined constraints. This motivates the need of exploring new architectures and new technologies; not only to maintain the economic benefit of scaling, but also to enable the solutions of emerging computer power and data storage hungry applications such as big-data and data-intensive applications. This paper discusses the emerging memristor device as complementary (or alternative) to CMOS device and shows how this device can enable new ways of computing that will at least solve the challenges of today's architectures for some applications. The paper shows not only the potential of memristor devices in enabling new memory technologies and new logic design styles, but also their potential in enabling memory intensive architectures as well as neuromorphic computing due to their unique properties such as the tight integration with CMOS and the ability to learn and adapt.