We intend to study, for the first time, the role that neutron star (NS) interior physics and gravitational wave (GW) torques have in the spin evolution of accreting millisecond X-ray pulsars (AMXPs). Accreting NSs, and AMXPs in particular, are one of the most interesting laboratories to study the physics of compact objects and a potentially interesting source of GWs. There is, however, currently a lively debate on the interpretation of apparent spin variations in these systems and without further theoretical work and a more solid understanding of the torque acting on the NS no progress can be made. NSs are a complex multi-component system, and it is well known that GW emission and the details of the interior physics will play a role in the spin evolution. However this effect has not been quantified, even though there are ongoing efforts to study the details of the magnetized accretion flow in the exterior. This is clearly a catastrophic omission. Through this project we intend to fill this scientific gap and aim to understand which GW emission mechanisms are compatible with the observed X-ray flux and pulse arrival time variability in AMXPs, and what kind of observable electromagnetic signature they would produce. We shall consider in detail the role that GW torques due to crustal asymmetries and unstable r-modes (fluid modes of oscillation governed by the Coriolis force) have on the spin evolution and study their signature in the quiescent thermal X-ray emission. Furthermore we shall study the effect that superfluidity, and the crust core coupling in particular, have on the NS’s response to a varying accretion torque in order to determine whether the proposed emission mechanisms (mountains and r-modes) are compatible with the observed fluctuations in the X-ray timing data. Such work is crucial if we are to interpret not only current but also future AMXP timing data and would allow us to build the kind of high-precision templates necessary for GW detection.