Abstract: On August 22, 1999, we completed the factorization of the 512--bit 155--digit number RSA--155 with the help of the Number Field Sieve factoring method (NFS). This is a new record for factoring general numbers. Moreover, 512--bit RSA keys are frequently used for the protection of electronic commerce---at least outside the USA---so this factorization represents a breakthrough in research on RSA--based systems. The previous record, factoring the 140--digit number RSA--140, was established on February 2, 1999, also with the help of NFS, by a subset of the team which factored RSA--155. The amount of computing time spent on RSA--155 was about 8400 MIPS years, roughly four times that needed for RSA--140; this is about half of what could be expected from a straightforward extrapolation of the computing time spent on factoring RSA--140 and about a quarter of what would be expected from a straightforward extrapolation from the computing time spent on RSA--130. The speed-up is due to a new polynomial selection method for NFS of Murphy and Montgomery which was applied for the first time to RSA--140 and now, with improvements, to RSA--155.