In air cooling of electronics, the heat transfer process from single chip packages mounted on circuit packs or boards is a combined mode process involving conduction within the chip carrier and the board and convection from the package and board surfaces. This chapter is devoted to a description of some selected conjugate problems in air cooling of electronic components. It discusses non-specialists to the subject by discussion of key features of conjugate problems involving electronic components on boards. The chapter illustrates the use of various approaches for conjugate analysis and to present physical insight that has recently been gained by their application in the author’s research group. It aims to broaden the scope of these conjugate treatments by pointing out the importance of conjugate phenomena in similar problems. The chapter presents analysis of and results from a limited class of conjugate sub-problems of the printed circuit board cooling problem, problems in which the flow is sufficiently simple.