This proceedings contains tutorials presented at the NATO Advanced Study Institute on Design and Implementation of Optimiz ation Software (Urbino, Italy, 20 June - 2 July, 1977) organized by the Committee on Algorithms (COAL) of the Mathematical Program ming Society. The authors are to be congratulated on their clear expositions plus their prompt cooperation. We were especially fortunate to have had two of the first pioneers in designing mathematical programming systems: W. Orchard-Hays and E. M. L. Beale. Surveying the contents the reader will find that the papers fall into three categories...