Summary: In this paper, at first, a lattice Boltzmann method for binary fluids, which is applicable at low viscosity values, is developed. The presented scheme is extension of the free-energy-based approach to a multi-relaxation-time collision model. Various benchmark problems such as the well-known Laplace law for stationary bubbles and capillary-wave test are conducted for validation. As an appealing application, instability of a rising bubble in an enclosed duct is studied and irregular behavior of the bubble is observed at very high Reynolds numbers. In order to highlight its capability to simulate high Reynolds number flows, which is a challenge for many other models, a typical wobbling bubble in the turbulent regime is simulated successfully. Then, in the context of phase-field modeling, a lattice Boltzmann method is proposed for multiphase flows with a density contrast. Unlike most of the previous models based on the phase-field theory, the proposed scheme not only tolerates very low viscosity values but also emerges as a promising method for investigation of two-phase flow problems with moderate density ratios. In addition to comparison to the kinetic-based model, the proposed approach is further verified by judging against the theoretical solutions and experimental data. Various case studies including the rising bubble, droplet splashing on a wet surface, and falling droplet are conducted to show the versatility of the presented lattice Boltzmann model.