Mammalia is a class of animal within the Phylum Chordata. Mammal classification has been through several iterations since Carolus Linnaeus initially defined the class. Many earlier ideas have been completely abandoned by modern taxonomists, among these are the idea that bats are related to birds or that humans represent a completely distinct group. Competing ideas about the relationships of mammal orders do persist and are currently in development. Most significantly in recent years, cladistic thinking has led to an effort to ensure that all taxonomic designations represent monophyletic groups. The field has also seen a recent surge in interest and modification due to the results of molecular phylogenetics.
George Gaylord Simpson's classic "Principles of Classification and a Classification of Mammals" (Simpson, 1945) was the original source for the taxonomy listed here. Simpson laid out a systematics of mammal origins and relationships that was universally taught until the end of the 20th century.
Since Simpson's 1945 classification, the paleontological record has been recalibrated, and the intervening years have seen much debate and progress concerning the theoretical underpinnings of systematization itself, partly through the new concept of cladistics. Though field work gradually made Simpson's classification outdated, it remained the closest thing to an official classification of mammals.
Read more about Mammal Classification: Molecular Classification of Mammals, Standardized Textbook Classification, McKenna/Bell Classification, Luo, Kielan-Jaworowska, and Cifelli Classification, Simplified Classification For Non-specialists