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One limitation of quantum chemical methods is the accuracy of the approximate many-body theoretical framework that is utilized. Accurate many-body formalisms for quantum chemical methods do exist, but these methods are computationally very expensive and have poor computational scaling with numbers of atoms (e.g. most scale as O(N5)). Methods also exist that are much less computationally expensive such as Hatree-Fock, Density Functional and the Hybrid Functional theories, but at a reduced representation of the exact many-body ground state. This severely limits either the system size that can be addressed accurately, or the accuracy of the representation of the many-body ground states. What is needed is a method that represents the many-body ground states accurately, but also with a low computational cost and computational scaling (ideally O(N)). We report on a methodology that achieves this goal, which we refer to as Many-Body Density Matrix Theory. This methodology is simple, scales linear with number of atoms with a computational cost similar to Hartree-Fock, size extensive, has no self-interaction, and is a variational upper bound to the many-body ground state energy. And additionally, this theory easily lends itself to the computation of highly accurate excited states