The electron g factor is an important fundamental structural parameter in atomic physics, as it reveals various mechanisms of interactions between electrons and external fields. Precise measurements of g factors of bound electrons in simple atomic and molecular systems provide an effective method for investigating the bound-state quantum electrodynamics (QED) theory. Especially in highly-charged heavy ions (HCIs), the strong electromagnetic interactions between the nuclei and inner-shell electrons provide unique opportunities to test QED under extremely strong fields. Accurate measurements of the g factors of the bound-state electrons are also important for determining nuclear effects, nuclear parameters and fundamental constants. The research on g factors of the bound-state electrons has become a frontier topic in fundamental physics. A Penning trap, which uses steady-state electromagnetic fields to confine charged particles, is utilized to precisely measure the g factor. This paper presents a comprehensive review of the experiments on g factors for few-electron simple systems in Penning traps, including experimental principles, experimental setups, measurement methods, and a summary of important research findings. The physical concept of the electron g factor and its historical research background are introduced. The electron g factor is considered as an effective probe to study higher-order QED effects. Through high-precision measurements of the free electron g factor, discrepancies between the fine-structure constants and other experimental results in atomic physics are identified. Notably, the g factor of the 1s electron in HCIs deviates significantly from the value for free electrons as the atomic number increases. Experimental principles, including the principle of the Penning trap and the principle of measuring the bound-state electron g factors are discussed. A double-trap experiment setup and related precision measurement techniques are also introduced.

This paper reviews several milestone experiments including (1) the stringent test of bound-state QED by precise measurement of bound-state electron g factor of a ¹¹⁸Sn⁴⁹⁺ ion, (2) measurement of the g factors of lithium-like and boron-like ions and their applications, and (3) measurement of the g-factor isotope shift by using an advanced two-ion balance technique in the Penning trap, providing an insight into the QED effects in nuclear recoil. Finally, this paper summarizes the challenges currently faced in measuring the g factors of bound-state electrons in few-electron ion systems and provides the prospects for the future developments of this field.