Many emerging communication applications transmit packets with small payloads, yet require rich metadata con for the proper functioning of the various involved protocol layers. Packet header compression becomes vital to efficiently support these emerging communication applications in resource-constrained environments, such as the Internet of Things (IoT). As a basis for the survey of the existing packet header compression standards and recent packet header compression research studies, we first introduce a novel set of five principles of packet header compression, namely the principles of identification, definition, placement, compression, and control. Based on these five principles, we survey the major header compression standards, spanning from the Thinwire protocol from 1984, via Robust Header Compression (RoHC) version 1 (2001) and version 2 (2008) to Static Con Header Compression (SCHC, 2020) and QPACK currently in draft status. We also comprehensively survey the recent header compression research studies in the past six years (2016-2021), following the introduced five principles as taxonomy. The survey of header compression research includes novel compression concepts and evaluations in various modern environments, such as in wireless mesh networks. We conclude the survey by outlining major open packet header compression research challenges along the five principles.