This paper focuses on the exploitation of linear-frequency-modulated continuous-wave (LFMCW) radars for noncontact range tracking of vital signs, e.g., respiration. Such short-range system combines hardware simplicity and tracking precision, thus outperforming other remote-sensing approaches in the addressed biomedical scenario. A rigorous mathematical analysis of the operating principle of the LFMCW radar in the context of vital-sign monitoring, which includes the explanation of key aspects for the maintenance of coherence, is detailed. A precise phase-based range-tracking algorithm is also presented. Exhaustive simulations are carried out to confirm the suitability and robustness against clutter, noise, and multiple scatterers of the proposed radar architecture, which is subsequently implemented at the prototype level. Moreover, live data from real experiments associated to a metal plate and breathing subjects are obtained and studied.