The authors have used a nine-parameter expanding source model that includes special relativity, quantum statistics, resonance decays, and freeze-out on a realistic hypersurface in spacetime to analyze in detail invariant [pi][sup +], K[sup +], and K[sup [minus]] one-particle multiplicity distributions and [pi][sup +] and [pi][sup [minus]] two-particle correlations in nearly central collisions of Pb + Pb at p[sub lab]/A = 158 GeV/c. These studies confirm an earlier conclusion for nearly central collisions of Si + Au at p[sub lab]/A = 14.6 GeV/c that the freeze-out temperature is less than 100 meV and that both the longitudinal and transverse collective velocities -- which are anti-correlated with the temperature -- are substantial. The authors also reconciled their current results with those of previous analyses that yielded a much higher freeze-out temperature of approximately 140 meV for both Pb + Pb collisions at p[sub lab]/A = 158 GeV/c and other reactions. One type of analysis was based upon the use of a heuristic equation that neglects relativity to extrapolate slope parameters to zero particle mass. Another type of analysis utilized a thermal model in which there was an accumulation of effects from several approximations. The future should witness the arrival of much new data on invariant one-particle multiplicity distributions and two-particle correlations as functions of bombarding energy and/or size of the colliding nuclei. The proper analysis of these data in terms of a realistic model could yield accurate values for the density, temperature, collective velocity, size, and other properties of the expanding matter as it freezes out into a collection of noninteracting hadrons. A sharp discontinuity in the value of one or more of these properties could conceivably be the long-awaited signal for the formation of a quark-gluon plasma or other new physics.