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In this talk, I will discuss recent results produced with co-authors Ivan Blank (KSU) and Brian Benson (UCR) regarding a formulation of the Mean Value Theorem for the Laplace-Beltrami operator on smooth Riemannian manifolds. We define the sets upon which mean values of (sub)-harmonic functions are computed via a particular obstacle problem in geodesic balls. I will thus begin by discussing the classical obstacle problem and then an intrinsic formulation on manifolds developed in our recent paper. After demonstrating how the theory of obstacle problems is leveraged to produce our Mean Value Theorem, I will discuss local and global theory for our family of mean value sets and potential connections between the properties of these sets and the geometry of the underlying manifold.