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This is a rough classification based on what professional mathematicians actually research and teach at the university level, along with some major elementary topics. Topics covered in primary and secondary school mathematics are listed at Math Wiki:Topics by level.

Foundations of mathematics
Logic and * mathematical logic
Set theory
Category theory
* Philosophy of mathematics
Number systems
Number theory
Elementary algebra
Abstract algebra
* Group theory
Linear algebra
* Matrix theory
* Universal algebra
Clifford algebra
Ordinary differential equations
Partial differential equations
* Integral equations
Real analysis
Complex analysis
* Functional analysis
* Variational methods
* Nonstandard analysis
* Measure theory
* Vector and tensor analysis
Graph theory
Euclidean geometry
Analytic geometry
Non-Euclidean geometry
* Modern geometry
* Algebraic geometry
Differential geometry
Applied mathematics
Discrete mathematics
* Actuarial mathematics
* Decision theory
* Game theory
* Mathematical modeling
Optimization or * Mathematical programming
Numerical analysis or * Numerical methods
* Scientific computing
* Theory of computation
Related disciplines
Recreational mathematics (including games and puzzles)
History of mathematics
* Mathematics education
* Information theory
* Systems theory
* Operations research
Mathematical physics