Abstract: Left-handed materials (LHM) have negative permittivity, negativepermeability, and negative refractive index. The propagation mechanism ofelectromagnetic waves in the LHMs is fundamentally different from that inconventional materials. This paper first discusses the intrinsicmechanisms for realizing negative permittivity and negative permeability,then the recent progress in the design and fabrication of LHMs issummarized. Various problems related to LHMs are discussed, including thedefinition of EM energy in an LHM, the retrieving method of effectiveparameters of LHMs, and the parameters influencing the relative bandwidthfor negative refraction of LHMs. The photon tunneling effect andelectromagnetic transparency phenomenon induced by LHMs are reviewed, theneutral inclusion concept in mechanics is particularlydiscussed for the transparency design. In the end, the physical meaning andrealization of acoustic LHMs are briefly introduced, the acoustictransparency phenomenon is discussed by analogy with the electromagneticcounterpart.