A numerical approach based on a diffuse-interface free energy lattice Boltzmann equation method is developed to gain fundamental insight in liquid-liquid dispersions. The approach relies on detailed resolution of the interaction of the dispersed and continuous phases at the microscopic level, including drop breakup and coalescence. Several studies have been performed. A study of the gravity-driven motion of a single n-butanol drop in water demonstrates that the method handles complex drop deformations, including shape-oscillating motion of drops. Simulations of a single liquid drop in simple shear flow were used to assess the impact of numerical parameters on drop deformation levels. At higher capillary numbers the simulations capture end-pinching and capillary wave breakup mechanisms. The method handles a range of shearing conditions from near-creeping flow, to drop Reynolds of 50, also a viscosity ratio range (dispersed phase over continuous phase viscosity) of 0.1−3.0. The feasibility of direct numerical simulations of turbulently agitated liquid-liquid dispersions is demonstrated. Three-dimensional simulations are carried out in fully-periodic cubic domains with grids of size 500³ and 1000³ and the resolution of the Kolmogorov length scale in the range 1 − 10 lattice units. The process of dispersion formation is visualized, revealing the details of breakup and coalescence. However, several numerical issues are encountered:​ appearance of spurious currents over liquid-liquid interface, dissolution of small drops, and easy coalescence of drops. The effects of each drawback on the results are discussed.