High precision measurements of the electron spin and orbital g-factors complement very well, the atomic/molecular experiments which test for parity, search for a permanent electric dipole moment, and investigate QED. In this talk, experimental evidence will be presented to show that the electron orbital g-factor is significantly anomalous. A search for an anomaly in the electron g-factors, particularly the spin g-factor, has provided stringent tests of QED (and therefore the Standard Model). However, it is currently assumed that the electron orbital g-factor is unaffected by the radiative interactions, though the anomaly (gS – 2) in the spin g-factor, is attributed to radiative corrections. It will be shown, both by experimental data and theoretical calculations, that the orbital g-factor is affected by radiative corrections. Furthermore, it is currently assumed, without the benefit of sufficient experimental investigations, that the electron has a uniform mass-to-charge distribution like a classical point particle, hence its orbital g-factor must be exactly equal to one, i.e, gL = 1. However, determinations from the measurement of the ratio of gJ values in In, Ga, Na, Ar, Ne and He, indicate that the anomaly in the electron orbital g-factor is of the order of 10-3 to 10-4 to very high precisions, and that the electron may not be a simple point particle. Alternatively, this could indicate the presence of hitherto unknown interactions. Thus, high-precision measurements of the electron g-orbital factors, will constitute also be a useful guide in the search for new physics beyond the Standard Model, while also providing a low-energy means of elucidating the nature or structure of the electron.