The Rockall Trough is one of the largest of the relatively unexplored basins that form the northeast Atlantic Passive Margin, and many aspects regarding its evolution remain unresolved. The basin can be subdivided into two distinct sub-basins termed the Northern and Southern Rockall Trough. The main period rifting occurred during the Cretaceous Period, with multiple rift events that culminated with the initiation of seafloor spreading in the North Atlantic Ocean. There are significant departures in the basin's subsidence-uplift history from classical post-rift thermal subsidence models} particularly during the Cenozoic evolution of the Rockall Trough. These include regional Paleogene uplift} associated with the development of the Iceland Plume} and a late Eocene deepening event. Possible explanations for the deepening event include compressional related buckling of the lithosphere or the loss of thermal support associated with the Iceland Plume. This study applies numerical} lithosphere-scale models to the Rockall Trough in order to gain insights into the mechanisms responsible for the early-evolution of the basin and the subsequent post-rift events. These models are used to test different hypotheses regarding the timing and nature of both extensional and compressional events} as well as the influence of thermal anomalies. The mechanisms that control the early evolution of the Rockall Trough have been modelled and include both uniform and non-uniform lithosphere extension combined with single- and multiple-rift scenarios. The formation of lower crustaI bodies, both magmatic underplating, mantle serpentinisation, and lithosphere rheology have been modelled in order to access the relative importance of all factors controlling the evolution of the Rockall Trough. In addition} the difference between the southern and northern Rockall Trough have been examined. The effects of the Iceland Plume have been modelled to assess its impact on basin evolution. This modelling has enabled the influence of thermal anomalies} the extrusion of large volumes of igneous material and the formation of magrnatic underplating to be investigated and quantified in terms of the uplift or subsidence generated. Model results show that compression during the Cenozoic Era can produce the rapid acceleration of subsidence observed in parts of the Rockall Trough. However, the magnitude of compressional related subsidence depends on the existing flexural deflection prior to compression. Consequently the interplay between the amount and distribution of crustaI thinning, the formation of lower crustal bodies, both magmatic underplating and mantle serpentinisation are crucial to accurately modelling the effects of compression on basin development.