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Despite continuous progress in X-ray angiography systems, X-ray coronary angiography is fundamentally limited by its 2D representation of moving coronary arterial trees, which can negatively impact assessment of coronary artery disease and guidance of percutaneous coronary intervention. To provide clinicians with 3D/3D+time information of coronary arteries, methods computing reconstructions of coronary arteries from X-ray angiography are required. Because of several aspects (e.g. cardiac and respiratory motion, type of X-ray system), reconstruction from X-ray coronary angiography has led to vast amount of research and it still remains as a challenging and dynamic research area. In the first part of this work, we review the state-of-the-art approaches on reconstruction of high-contrast coronary arteries from X-ray angiography. We mainly focus on the theoretical features in model-based (modelling) and tomographic reconstruction of coronary arteries, and discuss the evaluation strategies. We also discuss the potential role of reconstructions in clinical decision making and interventional guidance, and highlight areas for future research. In the second part, we look into the coronary artery reconstruction problem from a probabilistic perspective, and propose new algorithms for model-based 3D/3D+time reconstruction of coronary arteries. First, we formulate a novel probabilistic model-based centreline reconstruction method based on a Gaussian mixture model. Second, we propose a novel model-based 3D+time coronary artery centreline reconstruction method. The novelty of the method stems from the fact that it employs a spatiotemporal statistical model of the ventricular epicardium to cope with the ill-posedness of the reconstruction problem. Lastly, we extend our probabilistic 3D reconstruction method by using Student's t-distributions, and incorporating spatial regularisation and sparsity priors in a Bayesian framework. Thanks to these improvements, the reconstruction algorithm can handle cardiac motion inconsistencies between X-ray images due to finite gating accuracy, and noisy or erroneously segmented parts in 2D centreline segmentations.