This is the first of a pair of papers. It focuses on the development of the most notable phlogistic theories during the period 1766–1791, including the main experiments that their proponents proposed them to interpret. There was a rapid proliferation of late phlogistic theories, particularly from 1784, and the accounts of composition and important implications of the main theories are set out and their issues analysed. Each of them either reached impasses due to internal problems, or included features that made (...) them unacceptable even to other phlogistians. The expositions and analyses of these theories are given in terms of details that were in the literature at the time or otherwise potentially understandable by the participants given contemporary practice. Some relevant methodological aspects of the history of science are discussed, and the secondary literature is briefly surveyed. The second paper deals with the contemporary development of the new chemistry, and with theory comparison and theory choice in the same period. (shrink)