Very soft ferromagnets are commonly amorphous so that no magnetocrystalline anisotropy energy density contributes to the coercivity. This thesis focuses on a class of Fe-based alloys softer than amorphous ferromagnets but crystalline in structure, exhibiting linear, isotropic, as well as totally hysteresis-free magnetization. This class includes Fe-Ga, Al, Ge, Si and Pd alloys, and the majority of the experimental studies focused on FeGa alloys with compositions between 17 and 26 at. % Ga. FeGa has seen much research since the discovery of its large saturation magnetostriction, e­||-⊥, up to 310 ppm, reported by Clark et al. in 2000. Our studies probed the magnetic, magnetostrictive, and structural characteristics of these alloys to elucidate the origin of its anomalous magnetic and magnetoelastic properites. The magnetostriction we observe defies classical theories established by Joule in 1847, which pertain only single phase, crystalline materials. Magnetic anisotropy measurements demonstrate the FeGa alloys possess both cubic and uniaxial symmetry, indicating the presence of more than one phase, and a measured soft anisotropy constant of 1000 J/m3 for the cubic symmetry deviates from conventional proportionality between large magnetostriction and magnetic anisotropy for the majority of materials. Selected area diffraction and nanoelectron diffraction performed in a transmission electron microscope confirm the multi-phase nature of the FeGa alloys' microstructure, including disordered A2, ordered D03, and 6M D03 martensite phases. High resolution images show the microstructure is comprised of ~5 nm crystallites, even for alloys manufactured to be single-crystalline. Novel in situ field measurements were carried out in the microscope to probe the structure as a function of field, and these results demonstrate that the volume fraction of D03 appears to vary in response to the field. It is also shown that the magnetic and structural characteristics of FeGa alloys change with repeated cycles of thermal and magnetic measurements. The Fe82Ga18 alloy studied exhibited increased e­||-⊥ from 300 to 600 ppm, increased signal for uniaxial magnetic anisotropy, and increased D03 and 6M volume fraction. These results have signicant implications for future modelling of magnetostrictive behavior that takes into account varying phase content of multi-phase alloys, and the results also highlight the significance of processing and kinetics in the Fe-Ga system.