X-ray transients (XTs) provide some of the strongest evidence for the existence of stellar mass black holes (BHs). In these systems, X-ray outbursts occur over timescales of weeks to months and are followed by years of quiescence when the optical light from the system is dominated by the companion star. Successful programs for determining the orbit of the companion star, and thus the orbital parameters of the binary -- including the mass of the compact object -- have been carried out for the last three decades. These mass measurements have been used to explore the distribution of the stellar BH masses, which has important implications for the population and evolution of massive stars, the energetics and dynamics of supernova explosions, and the critical mass dividing neutron stars and black holes. However, both the small number of BH candidates and the systematic uncertainties in the determination of their masses, prevent us from extracting completely compelling statistical conclusions. In this paper we review the recent attempts to improve our understanding of the mass spectrum of collapsed stars, i.e., to measure their masses more accurately and to enlarge the sample of known quiescent black holes.