In this article, we discuss applications of our locally correlated lattice (LCL) theory to problems involving small molecules (e.g., simple alkanes), ionic liquids, and polymer melts and blends. The theory employs a compressible lattice model (segments and vacancies) combined with integral equation-based nearest neighbor segmentsegment probabilities (local correlations) leading to analytic solutions (e.g., a simple three-parameter equation of state) that are applicable for modeling the thermodynamic properties of fluids and fluid mixtures in both gas and liquid states of aggregation. The theory incorporates a physically meaningful set of molecular parameters that are transferable and that have been shown both to capture and to predict fundamental thermodynamic behavior. The model can be used to study the full spectrum of thermodynamic properties, ranging from pressurevolumetemperature behavior to phase equilibria for pure and mixed systems. The applications discussed in this paper, which encompass both a review of prior results as well as new work, involve extensive connections with experimental data as well as a new system for ranking polymers as a route to predictions regarding miscibility.