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.
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