  This paper is devoted to investigating the physically interesting optical and
electromagnetic properties, phenomena and effects of wave propagation in the
negative refractive index materials, which is often referred to as the {\it
left-handed media} in the literature. This paper covers a wide range of
subjects and related topics of left-handed media such as many mathematical
treatment of fundamental effects ({\it e.g.}, the reflection and the refraction
laws on the interface between LH and RH media, the group velocity and energy
density in dispersive materials, the negative optical refractive index
resulting from a moving regular medium, the reversal of Doppler effect in
left-handed media, the reversal of Cerenkov radiation in left-handed media, the
optical refractive index of massive particles and physical meanings of
left-handed media, the anti-shielding effect and negative temperature in
left-handed media, {\it etc.}), and their some applications to certain areas
({\it e.g.}, three kinds of compact thin subwavelength cavity resonators
(rectangular, cylindrical, spherical) made of left-handed media, the photon
geometric phases due to helicity inversions inside a periodical fiber made of
left-handed media, {\it etc.}).
