In this dissertation the analysis of the Stirling engine is presented, this research topic falls within the category of thermal energy conversion. The research that was conducted is presented in three chapters of which the topics are: the effects of allocation of volume on engine performance, the GPU-3 (Ground Power Unit - developed by GM) Stirling engine analysis, and the optimisation of a 1000 cm³ Stirling engine with finite heat capacity rates at the source and the sink. The Stirling engine has many advantages over other heat engines, as it is extremely quiet, has multi-fuel capabilities and is highly efficient. There is also significant interest in using Stirling engines in low to medium temperature solar thermal applications, and for waste heat recovery. To develop high-performance engines that are also economically viable, advanced mathematical models that accurately predict performance and give insight into the different loss mechanisms are required. This work aims to use and adapt such a model to analyse the effects of different engine parameters and to show how such a model can be used for engine optimisation using the Implicit Filtering algorithm. In the various analyses that are presented, the dynamic second order adiabatic numerical model is used and is coupled to equations that describe the heat and mass transfer in the engine. The analysis shows that the allocation of volume has a significant effect on engine performance. It is shown that in high-temperature difference (HTD) engines, increasing dead-volume ratio increases efficiency and decreases specific work output. In the case of low-temperature difference (LTD) and medium-temperature difference (MTD) engines, there is an optimal dead-volume ratio that gives maximum specific work output. It was also found that there are optimal swept volume ratios and that the allocation of heat exchanger volume has a negligible effect on engine performance - so long as the dead-volume ratio is optimal. The second order model with irreversibilities included was used to perform an exergy analysis of the GPU-3 Stirling engine. This model compared well with experimental results and the results from other models found in the literature. The results of the study show the two different approaches in modelling the engine losses and the effect that the various engine parameters have on the GPU-3 power output and efficiency. The optimisation of the 1000 cm³ Stirling engine was performed using a model with finite heat capacity rates at the source and the sink, fixed number of heater and cooler tubes, and four different regenerator mesh types. The engine geometry was optimised for maximum work output using the implicit filtering algorithm, and the results show the dominant effect that the regenerator has on engine performance and the geometry that gives maximum work output. The critical insights obtained from this research are the importance of the dead-volume ratio in engine analysis, the merits of the novel Second law Stirling engine model, and the importance of regenerator mesh choice and geometry. The Implicit filtering algorithm is also shown to be a suitable choice of optimisation algorithm to use with Stirling engine mathematical models.