The utilization of diagnostic imaging in the United States and worldwide is steadily growing. Due to a shortage of trained staff, the result is an increased and unsustainable workload for radiologists. Consequently, there is a high clinical need for the automation of cognitively challenging tasks, such as analyzing and interpreting medical images, to lighten the burden on radiologists and avoid a further increase in healthcare expenditure. Machine learning (ML), including deep learning (DL) offer a potential solution as these algorithms can learn to automatically recognize subtle patterns from large amounts of data and augment clinical decision-making. Despite the high enthusiasm for ML algorithms, concerns regarding their readiness for clinical deployment are impeding their clinical translation. In this thesis, we address three fundamental challenges to the translation of ML algorithms into clinical care settings. First, algorithms must perform robustly in routine clinical care settings. We demonstrate how appropriate image preprocessing improves the stability of handcrafted radiomic features extracted from brain MRIs. Second, the selected network design must be appropriate for a specific task. Here, we illustrate the advantages of shifting from a strictly discrete (ordinal) model of disease severity distribution to a continuously valued one. We introduce a generalized framework that can recover information lost by discretizing continuous variables into discrete training labels. Furthermore, disagreements in the labels generated by different annotators can be caused by individually varying decision thresholds. Therefore, we present the first design and demonstration of two methods that enable the joint learning of annotators' ordinal classification and their individual biases for a latent, continuously valued target variable like disease severity. Lastly, the performance of ML algorithms needs to be evaluated in a clinically meaningful manner. We address the disconnect between the subjective quality perception of clinical experts and the metrics that are typically used to evaluate performance. Furthermore, we identify criteria that experts use to evaluate the quality of automatically generated segmentations and describe their thought processes as they correct them. Based on the learnings from our work, we conclude with concrete recommendations for developing robust and trustworthy ML tools for medical imaging.