The interaction between communication entities, which is unique in the network setting, have been at the root of many difficult challenges in communications and information theory. This monograph advocates a sequential approach to make progress on the network communication problem. In order to do this, the channel (and source) model to capture the essence of the network communication problem are simplified and connected to the original problem. This leads to the concept of approximate characterizations of the channel capacity region and source coding rate regions. Approximate solutions to information theory problems are not new. However, they are by and far isolated results each with its own proof technique. This monograph describes a breakthrough systematic approach with two levels of approximation that can be applied to many problems. It concludes by demonstrating its application to four central problems in network information theory: (1) Relay networks, (2) Interference channels, (3) Multiple descriptions problem, and (4) Joint source-channel coding over networks. This monograph is intended for researchers and graduate students working at the forefront of research into network and communications problems.