Drugs are molecules, and molecules are naturally represented as graphs, with atoms as nodes and bonds as edges, making graph neural networks an intuitive computational framework for medicinal chemistry. The modern era began around 2017, when message-passing neural networks demonstrated that molecular representations could be learned directly from graph structures rather than relying on handcrafted features, marking a conceptual shift from encoding chemistry to enabling models to discover task-relevant chemical abstractions. Since then, the field has expanded to include attention-based models, geometric networks, equivariant architectures, and self-supervised molecular foundation models, which now impact property prediction, molecular design, synthesis planning, protein–ligand modeling, and drug repurposing. Despite this rapid progress, molecular graph learning still faces challenges such as data sparsity, noisy labels, uncertain generalization, and limited interpretability, issues that are particularly critical in drug discovery, where confident extrapolation is often more important than retrospective benchmark performance. Emerging directions focus on tighter integration with protein structure models, prospective validation in real discovery programs, and autonomous design–make–test–analyze systems, suggesting that the next phase will reward models that combine chemical insight, uncertainty awareness, and practical deployability. This narrative review traces the intellectual trajectory of molecular graph neural networks, linking early innovations in message passing to contemporary foundation models and highlighting their current and future roles in computational drug discovery.