Summary form only given. We describe a novel parallel algorithm that implements a dense matrix multiplication operation with algorithmic efficiency equivalent to that of Cannon's algorithm. It is suitable for clusters and scalable shared memory systems. The current approach differs from the other parallel matrix multiplication algorithms by the explicit use of shared memory and remote memory access (RMA) communication rather than message passing. The experimental results on clusters (IBM SP, Linux-Myrinet) and shared memory systems (SGI Altix, Cray XI) demonstrate consistent performance advantages over pdgemm from the ScaLAPACK/PBBLAS suite, the leading implementation of the parallel matrix multiplication algorithms used today. In the best case on the SGI Altix, the new algorithm performs 20 times better than pdgemm for a matrix size of 1000 on 128 processors. The impact of zero-copy nonblocking RMA communications and shared memory communication on matrix multiplication performance on clusters are investigated.