Multi-phase fluid flows are ubiquitous in natural phenomena and different industrial applications such as in food industry, the medical sector, heat exchangers, power generation systems, to name a few.  Understanding the underlining  physics of multi-phase flows  proved to be a challenging task due to presence of sophisticated dynamics, including the evolution of the interface between any pair of phases, thermodynamics and possibility of phase change, interactions between the fluid phases and a solid phase, etc.  Together with theoretical studies and experiments performed on a variety of multi-phase flow problems, numerical simulations have been employed by many researchers to scrutinise different aspects of the problem. During the last decades, a great many studies have been conducted aiming to provide more accurate numerical frameworks for investigating multi-phase flow problems.

Among the various complicated aspects of a multi-phase flow, the present thesis is focused on few characteristics of it the understanding of which requires more considerations and demands improvements in the numerical frameworks. First, we elaborate on the different interface tracking approaches suit the study of different multi-phase flows. In particular, a Volume of Fluid method, a compressible formulation of a diffuse interface approach, a Cahn-Hilliard phase field method, and an Immersed Boundary method are employed to study wetting phenomemna and fluxes at the interface. We have initially investigated biological-relevant membranes, extensional dynamics of a Elasto-viscoplastic material, and droplet spreading over rough surfaces.  In the second part of the thesis, we propose novel numerical methods and setups to investigate the phase change problems in both nanoscale and mesoscale. In particular, we developed a novel numerical method for the solidification problem, a pressure control setup for studying boiling at nanoscale, and a pressure based algorithm for modelling the boiling and evaporation.