The purpose of this paper is to present a systematic survey of the theoretical/experimental work accomplished in the area of digital phase-locked loops (DPLL's) during the period of 1960 to 1980. The DPLL represents the heart of the Building blocks required in the implementation of coherent (all digital) communications and tracking receivers. This survey is particularly motivated by the fact that microprocessor technology is advancing rapidly to the extent that sophisticated and flexible signal processing algorithms for communications and control can be realized in the digital domain. In fact, it is anticipated that the use of this signal processing technology will continue to expand rapidly in the development of advanced communications and tracking receivers, e.g., all digital modems. Consequently, one major purpose of this paper is to provide the reader with a survey and an overview of the theoretical and experimental work accomplished to date, thereby offering speedy access to the techniques and hardware developments which have been presented in a scattered literature. In addition, the authors feel that a tutorial article revealing the various theories, their relationships to one another, their shortcomings, their advantages and the assumptions on which each is based, would be of tremendous value to the engineer trying to decide what particular analysis procedure is applicable to his peculiar problem. Consequently, a byproduct of this presentation will be to point out unsolved problems of practical interest. A broad class of digital modulation techniques, viz. I-Q modulations and demodulation, are studied in a rather general way.