The main directions of studies of materials with negative index of refraction, also called left-handed or metamaterials, are reviewed. First, the physics of the phenomenon of negative refraction and the history of this scientific branch are outlined. Then recent results of studies of photonic crystals that exhibit negative refraction are discussed. In the third part numerical methods for the simulation of negative index material configurations and of metamaterials that exhibit negative index properties are presented. The advantages and the shortages of existing computer packages are analyzed. Finally, details of the fabrication of different kinds of metamaterials are given. This includes composite metamaterials, photonic crystals, and transmission line metamaterials for different wavelengths namely radio frequencies, microwaves, terahertz, infrared, and visible light. Furthermore, some examples of practical applications of metamaterials are presented.