Three-body systems provide the perfect framework for studying the quantum mechanics of both atoms and molecules. These studies can probe the fundamentals of particle interactions that underpin stability, reactivity and structure. This thesis contains a series of studies into the stability of ground state three-body systems. The focus of this thesis has been the high accuracy computation of threebody systems without recourse to either the Born-Oppenheimer (BO) approximation or approximation of the like-charged particle interaction, which for the case of atoms corresponds to the electron correlation. Principally the effects of mass and charge on the stability of systems is predicted. The complex nature of coupled electronic interaction is studied to the purpose of pursuing accurate electron correlation that underpins modern computational chemistry. The energies of three-body systems were calculated very accurately to typically mJmol−1 accuracy or better whilst still producing reliable wavefunctions of which all other properties of the system could be calculated accurately. The energies of some of these systems are the lowest to date and all use the latest finite masses as published by CODATA. Computational codes were developed to achieve this accuracy using both numerical and computer algebra methods. These were designed to be efficient, extendable and, importantly, to calculate highly accurate energies, expectation values and wavefunctions. The masses of any three particles in which there exists at least one bound state below the lowest continuum threshold were identified. The importance of symmetry breaking in a asymmetric system was made clear as the difference in the masses become larger. A new method was developed to identify the lowest charge of a nucleus that can bind two electrons. This method is more effective then those previously available as it produces a variational upper bound to the true minimum charge in a single calculation. The method was employed to identify the minimum nuclear charge required for binding two electrons in atoms of various nuclear masses. Additionally the electronic structure of such systems was investigated by a judicious partitioning that separates the two electrons into an inner and outer component relative to the nucleus. The electron correlation was calculated using the Löwdin definition and a highly accurate Hartree-Fock (HF) implementation specifically designed for the task. The effects this electron correlation has on various properties was quantified including the coulomb hole. A second coulomb hole was found which was previously thought to be an artefact but remains even with this highly accurate implementation.