Drug discovery is a complex process that involves multiple stages and tasks. However, existing molecular generative models can only tackle some of these tasks. We present Generalist Molecular generative model (GenMol), a versatile framework that uses only a single discrete diffusion model to handle diverse drug discovery scenarios. GenMol generates Sequential Attachment-based Fragment Embedding (SAFE) sequences through non-autoregressive bidirectional parallel decoding, thereby allowing the utilization of a molecular context that does not rely on the specific token ordering while having better sampling efficiency. GenMol uses fragments as basic building blocks for molecules and introduces fragment remasking, a strategy that optimizes molecules by regenerating masked fragments, enabling effective exploration of chemical space. We further propose molecular context guidance (MCG), a guidance method tailored for masked discrete diffusion of GenMol. GenMol significantly outperforms the previous GPT-based model in de novo generation and fragment-constrained generation, and achieves state-of-the-art performance in goal-directed hit generation and lead optimization. These results demonstrate that GenMol can tackle a wide range of drug discovery tasks, providing a unified and versatile approach for molecular design. Our code is available at https://github.com/NVIDIA-Digital-Bio/genmol.