The focus of this monograph is to explore game theoretic modeling and mechanism design for problem solving in Internet and network economics. The monograph is structured into two parts. The first part (Chapter 2) contains an overview of foundational concepts and key results in mechanism design. This chapter is intended as a self-sufficient introduction to mechanism design theory with the help of numerous stylized examples from network economics. The second part of the monograph contains an exposition of representative game theoretic problems in three different network economics situations and a systematic exploration of mechanism design solutions to these problems. This part has three chapters: Chapter 3 deals with the sponsored search auction problem, Chapter 4 with the resource allocation problem in computational grids, and Chapter 5 with the robust broadcast protocol design problem in ad hoc networks. The monograph concludes with Chapter 6 which provides several pointers to the relevant literature to facilitate a deeper and broader investigation of problem solving with mechanism design.