We exploit the one-to-one correspondences between the recursive least-squares (RLS) and Kalman variables to formulate extended forms of the RLS algorithm. Two particular forms of the extended RLS algorithm are considered: one pertaining to a system identification problem and the other pertaining to the tracking of a chirped sinusoid in additive noise. For both of these applications, experiments are presented that demonstrate the tracking superiority of the extended RLS algorithms compared with the standard RLS and least-mean-squares (LMS) algorithms.