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This thesis deals with the systematic treatment of quantum-mechanical systemsin post-Newtonian gravitational fields. Starting from clearly spelled-outassumptions, employing a framework of geometric background structures definingthe notion of a post-Newtonian expansion, our systematic approach allows toproperly derive the post-Newtonian coupling of quantum-mechanical systems togravity based on first principles. This sets it apart from more heuristicapproaches that are commonly employed, for example, in the description ofquantum-optical experiments under gravity.  Regarding single particles, we compare simple canonical quantisation of afree particle in curved spacetime to formal expansions of the minimally coupledKlein-Gordon equation, which may be motivated from QFT in curved spacetimes.Specifically, we develop a general WKB-like post-Newtonian expansion of the KGequation to arbitrary order in $c^{-1}$. Furthermore, for stationaryspacetimes, we show that the Hamiltonians arising from expansions of the KGequation and from canonical quantisation agree up to linear order in particlemomentum, independent of any expansion in $c^{-1}$.  Concerning composite systems, we perform a fully detailed systematicderivation of the first order post-Newtonian quantum Hamiltonian describing thedynamics of an electromagnetically bound two-particle system situated inexternal electromagnetic and gravitational fields, the latter being describedby the Eddington-Robertson PPN metric.  In the last, independent part of the thesis, we prove two uniqueness resultscharacterising the Newton--Wigner position observable for Poincar\'e-invariantclassical Hamiltonian systems: one is a direct classical analogue of thequantum Newton--Wigner theorem, and the other clarifies the geometricinterpretation of the Newton--Wigner position as `centre of spin', as proposedby Fleming in 1965.

Post-Newtonian Description of Quantum Systems in Gravitational Fields