Space Domain Awareness (SDA) is concerned with space objects and their interactions in the space environment in temporal and spatial dimensions. The amount of SDA data to be processed is rapidly increasing as SDA sensors improve and resident space objects (RSO) proliferate. Recently, data-driven methods are being proposed for efficiently pro- cessing large amounts of data, reducing the complexity and workload associated with SDA activities, and to build and maintain space catalogues with more realistic space object characteristics and orbital elements. The primary objective of this research is to develop and validate a set of robust and gen- eralisable overarching machine learning models for two inverse problems in SDA, namely TLE estimation and light curve inversion for HAMR objects. This thesis is the first to investigate the feasibility of applying machine learning to predict TLEs from time-series osculating orbital elements. It is also the first to apply proba- bilistic deep generative neural networks to predict the physical characteristics of HAMR objects from light curves. A 6DoF space object simulator software suite for complex- shaped objects and Monte-Carlo based two line element (TLE) estimation programme were developed to achieve this. The key finding was that machine learning can be used to predict physical parameters such as TLEs and shapes, sizes, and attitude rates of complex- shaped HAMR objects. Secondly, the results showed that probabilistic deep generative neural networks are better than neural networks and gradient boosting trees when there are epistemic and aleatoric uncertainties in the data. The predictions of probabilistic deep neural network ensembles are physically more bounded. This has demonstrated that the proposed overarching machine learning models can reduce the complexity and workload associated with SDA activities, such as building and maintaining space catalogues with more realistic space object characteristics and orbital elements by efficiently processing large amounts of data.