This is a rough classification based on what professional mathematicians actually research and teach at the university level, along with some major elementary topics.
- Foundations of mathematics
- Logic
- Set theory
- Category theory
- Philosophy of mathematics
- Arithmetic
- Algebra
- Elementary algebra
- Abstract algebra
- Linear algebra
- Universal algebra
- Analysis
- Real analysis
- Complex analysis
- Functional analysis
- Variational methods
- Nonstandard analysis
- Graph theory
- Geometry
- Applied mathematics
- Numerical analysis
- Combinatorics
- Probability
- Statistics
- Decision theory
- Game theory
- Mathematical modeling
- Theory of computation
- Related disciplines
- Recreational mathematics (including games and puzzles)
- History of mathematics
- Mathematics education
- Information theory
- Systems theory
- Operations research
- Mathematical physics