Nonrandom hydrogen bonding (NRHB) lattice theory is extended here to multicomponent fluid mixtures rigorously. The model accounts for nonideal thermodynamic behavior of mixtures due to molecular connectivity of nonspherical molecules, weak van der Waals forces between first-neighbor molecular segments, and hydrogen-bonding interactions. The random distribution of molecules in the lattice is calculated using the generalized Staverman theory, while for the nonrandom correction, the Guggenheim quasi-chemical theory is adopted. Finally, the hydrogen-bonding contribution is based on Veytsman statistics as implemented in lattice fluid theory by Panayiotou and Sanchez. The equation of state is coupled with the mass action law due to hydrogen bonding, and both are solved simultaneously. The model is applied for the calculation of vapor-liquid, liquid-liquid, and vapor-liquid-liquid equilibria at low and high pressures of binary mixtures of fluids with large molecular size differences and/or different types of interactions between unlike molecules. In addition, the model is applied to correlate low-pressure polymer-solvent data. Good agreement between experimental data and model predictions/correlations is obtained in all cases. Comparisons against mixture predictions from the model ignoring nonrandom contributions and from lattice-fluid-hydrogen-bonding model show a clear improvement of the NRHB model.