We present here a brief overview of current-induced spin polarization in bulk semiconductors and semiconductor structures of various dimension. The role of band structure and spin relaxation processes is discussed. The related phenomena, such as spin Hall effect, inverse spin Hall effect and other are discussed. Our recent results in this field are presented as well.Spin-orbit coupling (SOC) is a relativistic effect that provides a link between spin and electric field (including the field of light wave). The SOC is the basis of modern concept of semiconductor spintronics. One can distinguish two main type of the spin orientation at current carrying through the sample: (i) spin Hall effect (SHE), which is the spatial separation of carriers with opposite spins and (ii) homogeneous in the sample polarization. This paper is focused mainly on effect of homogeneous current-induced spin polarization (CISP), however, the related phenomena such as SHE, inverse SHE and other are discussed as well. We will conduct presentation adhering to the chronology of events, which coincides with the transition from bulk semiconductors to low-dimensional ones.SHE is due to so-called Mott-scattering [1], known in atomic physics, that is the asymmetry of the scattering relative to the plane determined by momentum and spin that in turn is due to SOC. In semiconductors the role of SOC increases and such an effect is several order of magnitude stronger. In 1971 Dyakonov and Perel predicted this phenomenon in semiconductors [2,3]. The name SHE was introduced later by Hirsch in 1999 [4] and the fist experimental observation of SHE was in Awschalom group [5] through more than 30 years after the prediction. The qualitative picture of SHE is depicted in Fig. 1a, where one can see, that the electrons with opposite spin orientation experience the scattering predominantly in opposite directions. This leads to accumulation of carriers with opposite spin projection at opposite sides of the sample at the length scales about spin diffusion length l s .The inverse SHE, i.e. the appearance of the dc current due to the nonhomogeneous spin polarization, was predicted by Averkiev and Dyakonov in 1983 [6] and detected by Bakun et al. [7] in the case, when inhomogeneous spin polarization appears due to spin diffusion after interband excitation under condition of optical orientation.There are two main differences between CISP and SHE: (i) homogeneous vs non-homogeneous spin polarization; (ii) CISP takes place only in gyrotropic semicon- * Electronic address: kokurinia@math.mrsu.ru ductors, while SHE can be realized even in structure with inversion center.Phenomenologically CISP is described by second rank pseudotensorQ linking the current density vector j and average spin pseudovector SFrom the symmetry point of view the gyrotropy means that the polar vector (current density) and axial vector (spin) transform under the same irreducible representation. Phenomenological description (1) shows that pseudotensorQ has non-zero component Q ik only if S i and j k equ...