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We review the existing and present the new results of $\bf kp$ calculations of the electron, hole, and exciton effective $g$-factors in semiconductor nanocrystals of different shape and symmetry. We propose a simple yet accurate method for calculation of electron $g$-factor size dependence in bare nanocrystals within the eight-band Kane model. Using the spherical approximation for Luttinger Hamiltonian we find the dependence of hole $g$-factor on light to heavy hole effective mass ratio in semiconductor nanostructures with spherical, axial, and cubically symmetric shape. We show that the non-equidistant Zeeman splitting of the four-fold degenerate hole state may take place in cube and spheroidal nanocrystals. We present a comparison of the calculated hole $g$-factors in nanostructures based on II-VI and III-V semiconductors for different sets of the Luttinger parameters and analyze the main effects contributing to the $g$-factor renormalization with the respect to the bulk value. We discuss different approaches to the definition of the hole and exciton $g$-factors which should be taken into account during the analysis of the experimental data and compare our results of $g$-factor calculations with the experimental data for semiconductor spherical nanocrystals and thin nanoplatelets available in the literature.