We investigate theoretically the Zeeman effect on the lowest confined electron in quantum wires and quantum dots. A general relation is established between the symmetry of a low-dimensional system and properties of the electron g factor tensor, g . The powerful method used earlier to calculate the transverse g factor in quantum wells is extended to one-dimensional (1D) and 0D zinc-blende-based nanostructures and analytical expressions are derived in the frame of Kane's model for the g factors in quantum wells, cylindrical wires, and spherical dots. The role of dimensionality is illustrated on two particular heteropairs, GaAs/AlGaAs and GaInAs/InP. The efficiency of the developed theoretical concept is demonstrated by calculating the three principal values of the g factor tensor in rectangular quantum wires in dependence on the wire width to establish also the connection with the 2D case.