This paper introduces and develops Möbius homology, a homology theory for representations of finite posets into abelian categories. Although the connection between poset topology and Möbius functions is classical, we go further by establishing a direct connection between poset topology and Möbius inversions. In particular, we show that Möbius homology categorifies the Möbius inversion, as its Euler characteristic coincides with the Möbius inversion applied to the dimension function of the representation. We also present a homological version of Rota's Galois Connection Theorem, relating the Möbius homologies of two posets connected by a Galois connection. Our main application concerns persistent homology over general posets. We prove that, under a suitable definition, the persistence diagram arises as an Euler characteristic over a poset of intervals, and thus Möbius homology provides a categorification of the persistence diagram. This furnishes a new invariant for persistent homology over arbitrary finite posets. Finally, leveraging our homological variant of Rota's Galois Connection Theorem, we establish several results about the persistence diagram.