We present a comprehensive study of the observational constraints on spatially flat cosmological models containing a mixture of matter and quintessence—a time-varying, spatially inhomogeneous component of the energy density of the universe with negative pressure. Our study also includes the limiting case of a cosmological constant. We classify the observational constraints by redshift: low-redshift constraints include the Hubble parameter, baryon fraction, cluster abundance, the age of the universe, bulk velocity and the shape of the mass power spectrum; intermediate-redshift constraints are due to probes of the redshift-luminosity distance based on Type Ia supernovae, gravitational lensing, the Lyα forest, and the evolution of large-scale structure; high-redshift constraints are based on measurements of the cosmic microwave background temperature anisotropy. Mindful of systematic errors, we adopt a conservative approach in applying these observational constraints. We determine that the range of quintessence models in which the ratio of the matter density to the critical density is 0.2 Ω_m 0.5, and the effective, density-averaged equation of state is -1 ≤ w -0.2, is consistent with the most reliable, current low-redshift and microwave background observations at the 2 σ level. Factoring in the constraint due to the recent measurements of Type Ia supernovae, the range for the equation of state is reduced to -1 ≤ w -0.4, where this range represents models consistent with each observational constraint at the 2 σ level or better (concordance analysis). A combined maximum likelihood analysis suggests a smaller range, -1 ≤ w -0.6. We find that the best-fit and best-motivated quintessence models lie near Ω_m ≈ 0.33, h ≈ 0.65, and spectral index n_s = 1, with an effective equation of state w ≈ -0.65 for "tracker" quintessence and w = -1 for "creeper" quintessence.