The emergence of the Internet of Things (IoT) has enabled billions of devices that collect large amounts of data to be connected. Therefore, IoT security has fundamental requirements. One critical aspect of IoT security is data integrity. Cryptographic hash functions are cryptographic primitives that provide data integrity services. However, due to the limitations of IoT devices, existing cryptographic hash functions are not suitable for all IoT environments. As a result, researchers have proposed various lightweight cryptographic hash function algorithms. In this paper, we discuss advanced lightweight cryptographic hash functions for highly constrained devices, categorize design trends, analyze cryptographic aspects and cryptanalytic attacks, and present a comparative analysis of different hardware and software implementations. In the final section of this paper, we highlight present research challenges and suggest future research topics related to the design of lightweight cryptographic hash functions.