We present a new approach to fault tolerance for High Performance Computing system. Our approach is based on a careful adaptation of the Algorithmic Based Fault Tolerance technique (Huang and Abraham, 1984) to the need of parallel distributed computation. We obtain a strongly scalable mechanism for fault tolerance. We can also detect and correct errors (bit-flip) on the fly of a computation. To assess the viability of our approach, we have developed a fault tolerant matrix-matrix multiplication subroutine and we propose some models to predict its running time. Our parallel fault-tolerant matrix-matrix multiplication scores 1.4 TFLOPS on 484 processors (cluster jacquard.nersc.gov) and returns a correct result while one process failure has happened. This represents 65% of the machine peak efficiency and less than 12% overhead with respect to the fastest failure-free implementation. We predict (and have observed) that, as we increase the processor count, the overhead of the fault tolerance drops significantly.