We have theoretically shown that optical properties of semiconductor nanorods are controlled by 1D excitons. The theory, which takes into account anisotropy of spacial and dielectric confinement, describes size dependence of interband optical transitions, exciton binding energies. We have demonstrated that the fine structure of the ground exciton state explains the linear polarization of photoluminescence. Our results are in good agreement with the measurements in CdSe nanorods.PACS numbers: 71.70.Gm, 77.22.Ej There is growing interest in nano-size crystalline semiconductor structures of various shapes such as nanocrystals (NCs) [1], nanorods (NRs) [2] and nanowires (NWs) [3] created by the "from the bottom up" technological approach. Size-tunable control of their optical and transport properties combined with the ability to move these nano-size objects around with precise control opens the exciting possibilities for the creation of new functional materials which can be used in unlimited applications. Among these nanostructures, the NCs are the most heavily studied and one can find a broad description of their properties and their potential applications in the reviews of Brus [4] and Alivisatos [5].The optical properties of NRs, however, differ significantly from those of NCs. Compared to NCs, the NRs show higher photoluminescence (PL) quantum efficiency [2], strongly linear polarized PL [6,7], an increase of the global Stokes shift [6], and significantly faster carrier relaxation [8]. The Auger processes in NRs are strongly suppressed relative to those in NCs [9,10], which subsequently decreases the optical pumping threshold for stimulated emission [10,11]. The size and shape dependence of the optical and tunneling gaps measured in CdSe NRs [12] shows an unexpectedly large difference that cannot be explained by the electron-hole Coulomb correction to the optical gap used for NCs. To describe their measurements Katz et al.[12] applied the four-band theory developed by Sercel and Vahala [13]. The most important result of the latter paper was the prediction of an inverse order of light and heavy hole subbands in NWs, which was later confirmed by semiempirical pseudopotential calculations [6,14,15] and numerical calculations within the 6 band model [16]. An anisotropy of the light hole-toelectron optical transition matrix element predicts some degree of the PL linear polarization parallel to the NW [17,18].The dielectric confinement connected with the difference between dielectric constants of semiconductor crystallites, κ s , and surrounding medium, κ m , (for example, see [19]) usually does not affect the optical spectra of NCs because the charge distributions of an excited electron and hole practically compensate each other at each point of the NC. This leads to a complete screening of an electric field of the total charge from penetration into the surrounding medium. This is not the case, however, in NRs where the electron and hole are at a distance larger than a NR radius and interact predominantly throu...