Chapters have been updated and extended and number of examples are given for better understanding. To balance the character of a textbook with the dynamic nature of this research, many useful references have been added to develop a deeper understanding for the interested reader. Theoretical advances have been made in many directions. Application of Vague algebra which is generalization of fuzzy algebra has advanced in a variety of ways and in many disciplines. This vague technology can be found in artificial intelligence, computer science, decision making, and control engineering. This textbook updated the research agenda with chapters on vague group and vague normal groups, vague products, vague homomorphisms and vague weights, vague universal algebras,vague graphs and vague linear programming problem. A study of Vague groups, Vague universal algebras and vague graphs is to provide a textbook for courses in vague algebra and a book that can be used as an introduction.