Two kinds of new square-root algorithms for Rauch-Tung-Striebel (RTS) smoothing formulas are suggested : stable square-root algorithms and a fast square-root (or so-called Chandrasekhar-type) algorithm. The new stable square-root algorithms use square-root arrays composed of smoothed estimates and their error covariances. These square-root algorithms provide many advantages over the conventional algorithms with respect to systolic array and parallel implementations as well as numerical stability and conditioning. For the case of constant-parameter systems, a fast square-root algorithm is suggested, which requires less computation than others.