Symbolic computation and computer algebra systems are usually known to be either very slow, or memory expensive. However, some specific symbolic computation problems have received in the last years new algorithmic solutions, which enabled to push further the limits of what is doable within a reasonable amount of time and space. Some noticeable examples are polynomial factorisation, lattice reduction, Groebner basis computation. We will present a few such algorithms, together with a state-of-the-art of what problems computer algebra systems can (or cannot) solve, and for each problem what the current frontiers are.