Abstract: This talk will be focused on the theory of noncommutative Einstein manifolds. I will recall the idea of a spectral triple and specialize to the case that requires a real structure J to exist in this spectral triple. We can think of this real structure as a simultaneous generalization of the charge conjugation operator acting on the spinor bundle over a spin manifold, and as the Tomita J-operator of Tomita-Takesaki theory that classifies modular automorphisms of von Neumann algebras. I will then define a noncommutative Einstein manifold in terms of a spectral characterization -- more precisely, in terms of the asymptotic expansion of its heat trace. We will see that this definition is, in fact, correct, because of the equivalence to the classical notion of an Einstein manifold in the commutative case. Furthermore, I will explain the noncommutative Einstein-Hilbert action and Connes' trace theorem, which connects Dirac operators and the Einstein-Hilbert action via the Wodzicki residue. Finally, I will introduce the fundamental examples of noncommutative Einstein manifolds and explain how we classify abstract noncommutative spin geometries on manifolds via Connes' reconstruction theorem.