Multi-layered, golf balls having a core made of a foamed composition are provided. The core preferably has a foam inner core (center) and surrounding thermoset or thermoplastic outer core layer. The core also includes a structural insert such as, for example, a solid shell, perforated shell, a lattice or mesh, or a central hub with extending arms, and the like. The core layers have different hardness gradients and specific gravity values. The ball further includes a cover having at least one layer. By adjusting the respective specific gravity values of the core layers, the Moment of Inertia (MOI) of the ball can be increased or decreased. In one embodiment, the ball has generally good flight distance and low spin rate.