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The key to evaluating the formation history and evolution of the Moon lies in understanding the current state of its interior. We used a multidisciplinary approach to explore the current day lunar structure and composition with the aim of identifying signatures of formation and early evolution. We constructed a large number of 1D lunar interior models to explore a wide range of potential structures and identified those models that match the present day mass, moment of inertia, and bulk silicate composition of the Moon. In an advance on previous studies, we explicitly calculate the physical and elastic properties of the varying mineral assemblages in the lunar interior using multicomponent equations of state. We considered models with either a compositionally homogeneous mantle or a stratified mantle that preserved remnants of magma ocean crystallization, and tested thermal profiles that span the range of proposed selenotherms. For the models that reproduced the observed mass and moment of inertia, we found a narrow range of possible metallic (iron) core radii (269-387 km) consistent with previous determinations. We explored the possibility of an ilmenite bearing layer both below the crust and at the core-mantle boundary as a potential tracer of magma ocean solidification and overturn. We observed a trade-off between the mass of the upper and lower ilmenite-bearing layers and structures that have undergone mantle overturn are both consistent with present observations. Plain Language Summary In order to understand how the Moon formed, along with the following history including the processes that change and shape it, the current state of …