  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.
