  Negative index materials are artificial structures whose refractive index has
a negative value over some frequency range. These materials were postulated and
investigated theoretically by Veselago in 1964 and were confirmed
experimentally by Shelby, Smith, and Schultz in 2001. New fabrication
techniques now allow for the construction of negative index materials at scales
that are interesting for applications, which has made them a very active topic
of investigation. In this paper, we report various mathematical results on the
properties of negative index materials and their applications. The topics
discussed herein include superlensing using complementary media, cloaking using
complementary media, cloaking an object via anomalous localized resonance, and
the well-posedness and the finite speed propagation in media consisting of
dispersive metamaterials. Some of the results have been refined and have
simpler proofs than the original ones.
