Problem-solving in the undergraduate curriculum typically occurs in content-focused courses that emphasize applying a conceptual and mathematical understanding of key physics principles to given situations. This project expands the notion of problem-solving by characterizing the breadth of problem-solving activities carried out by graduate students in physics-intensive research. In 10 in-depth interviews, PhD students were asked to describe routine, difficult, and important problems they engage in. A grounded theory analysis resulted in a framework with three dimensions: problem context (e.g., experiments, software, or math), activity (e.g., design or troubleshooting), and feature that made the problem hard (e.g., complexity or insufficient resources). Problem contexts usually extended beyond theory and mathematics (e.g., experiments, data analysis, and computation). Important problem contexts blended soft and technical skills (e.g., communication and collaboration). Routine problem activities tended to be well-defined (e.g., troubleshooting) while important ones were more open-ended and had multiple solution paths (e.g., evaluating options). The results can inform curriculum development and PER with an expanded view of problem-solving.