The first book on ABS methods "ABS-Projection Algorithms: Mathematical Techniques for Linear and Nonlinear Algebraic Equations" by J. Abaffy and E. Spedicato was published by Ellis Horwood in 1989. This book was then translated to Russian and Chinese. Chinese researchers made several important applications of ABS methods to optimization and professors Zhang Liwei, Xia Zunquan and Feng Enmin published a monograph on these results under the title "Introduction to ABS methods in Optimization" in Chinese by the Dalian University of Technology Press in 1999. Considering the great number of results and application we feel that it is time to summarize the most important results including the recent ones in two volumes whose subject is the application of ABS methods for solving nonlinear system of equations, twice is enough method for conjugate directions and optimization. It can be used as a second course for advanced master, or PhD students in fields such as operation research, linear and nonlinear algebra. The readers must have a basic knowledge of linear and nonlinear algebra and mathematical programming.