The Level 3 BLAS (BLAS3) are a set of specifications of Fortran 77 subprograms for carrying out matrix multiplications and the solution of triangular systems with multiple right-hand sides. They are intended to provide efficient and portable building blocks for linear algebra algorithms on high performance computers. We describe algorithms for the BLAS3 operations that are asymptotically faster than the conventional ones. These algorithms are based on Strassen's method for fast matrix multiplication, which is now recognized to be a practically useful technique once matrix dimensions exceed about 100. We pay particular attention to the numerical stability of these "fast BLAS3". Error bounds are given and their significance is explained and illustrated with the aid of numerical experiments. Our conclusion is that fast BLAS3, although not as strongly stable as conventional implementations, are stable enough to be suitable for many applications.