Present and future supercomputers offer many opportunities and ad vantages to attack complex and demanding industrial and applied mathematical problems, but provide also new challenges. In the Peta Flops regime, these concern both, the way to exploit the increasingly available power and the need of designing algorithms which are scal able and fault-tolerant at the same time. An example of a probabilistic domain decomposition method, which is indeed scalable and naturally fault-tolerant, is presented. Grid computing should also be mentioned as an increasingly popular way to perform massively distributed com puting: it represents a way to exploit computing power, aside the ex isting supercomputers. Beyond classical supercomputers there is the prospective quantum computer, in view of which it is advisable to start now a search for suitable algorithms for entire classes of problems. Due to the ever more challenging problems put forth by the needs of scientific computing, computing power increases continuously, making it possible to cope with more complex applications. These problems include both, purely scientific advances and very practical industrial needs.