  A five-level {\Pp} model of the band structure for GaAs-type semiconductors
is used to describe the spin $g^*$-factor and the cyclotron mass $m^*_c$ of
conduction electrons in GaAs/Ga$_{1-x}$Al$_x$As quantum wells in an external
magnetic field parallel to the growth direction. It is demonstrated that the
previous theory of the $g^*$-factor in heterostructures is inadequate. Our
approach is based on an iteration procedure of solving 14 coupled differential
{\Pp} equations. The applicability of the iteration procedure is verified. The
final eigenenergy problem for the conduction subbands is reduced to two
differential equations for the spin-up and spin-down states of consecutive
Landau levels. It is shown that the bulk inversion asymmetry of III-V compounds
is of importance for the spin $g^*$-factor. Our theory with no adjustable
parameters gives an excellent description of experimental data on the electron
spin $g^*$-factor in GaAs/Ga$_{0.67}$Al$_{0.33}$As rectangular quantum wells
for different well widths between 3 and 12 nm. The same theory describes very
well experimental cyclotron masses in GaAs/Ga$_{0.74}$Al$_{0.26}$As quantum
wells for the well widths between 6 and 37 nm.
