Molecular machine learning techniques have recently shown great promise for important computational drug discovery tasks such as molecular property prediction and activity cliff prediction. The success of such methods, however, crucially depends on the way in which molecular compounds are transformed into informative feature vectors that can be fed into a machine learning pipeline. This is referred to as the problem of molecular representation. In my DPhil project, I am investigating the potential of modern graph-based molecular representation techniques to outperform classical molecular representations such as structural fingerprints and physicochemical descriptor vectors. I am particularly interested in developing novel self-supervised learning strategies for graph neural networks operating on molecular graphs, identifying and removing hidden performance barriers of state-of-the-art molecular representation methods, and using the gained insights to design new tailored deep learning architectures for molecular property prediction and activity cliff prediction.