Here are some links.
- Axiom and Maxima are good general purpose computer algebra systems.
- DataMelt is a free Java-based math software with a lot of examples
- GAP is a system for computational discrete algebra (with particular emphasis on computational group theory).
- PARI/GP is a CAS for fast computations in number theory.
- SAGE is a kind of unified framework for several systems, including GAP, PARI, and Maxima.
- Octave is a system for numerical computations (it is close to Matlab).
- Cadabra is a computer algebra system designed for the solution of the field theory problems.
- CoCoA stands for "Computations in commutative algebra".
- KANT / KASH stands for "Computational Algebraic Number Theory".
- Macaulay 2 is a system for research in commutative algebra and algebraic geometry.
- Snap is a computer program for studying arithmetic invariants of hyperbolic 3-manifolds.
- Symmetrica is an object oriented computer algebra system for representations, combinatorics and applications of symmetric groups.