In this paper we show that the order recursive least squares (LS) algorithms can be systematically studied as follows: (1) Determine various estimator structures (connection of basic building cells) based on the known properties of the input data vector and required output, and (2) Investigate possible realizations of time-updating and their properties implemented in the elementary cells. In view of this approach, we show that using a structure from (1) in combination with a time-update realization from (2) almost always forms a valid order-recursive LS (ORLS) algorithm. All known ORLS algorithms can be derived under a unified framework by using this approach. The properties of ORLS algorithms, such as computational complexity and sensitivity to the round-off error, can also be investigated in a systematic manner. This method not only simplifies the investigation of existing algorithms, but it also is powerful enough for creating new ORLS algorithms. These points will become clearer during the course of the development of this paper.