The phase-field method, also known as the diffuse interface method, has gained popularity in simulating interfacial flows. The interface between two immiscible fluids is treated as a diffuse layer governed by a phase-field variable that obeys the Cahn-Hilliard equation. In my talk, I will first describe two recent contributions to this class of methods: a generalization of the theoretical framework to account for complex rheology of non-Newtonian fluids, and a finite-element implementation with adaptive meshing for highly accurate interfacial resolution. These have allowed us to simulate interfacial dynamics of complex fluids with a refined understanding of the microscopic physics. Then I will discuss numerical simulations of the unique partial coalescence process. This refers to the phenomenon that a drop falling onto a fluid-fluid interface does not merge with it completely but leaves a smaller daughter drop behind, thus forming a cascade of partial coalescence. With a wide range of length scales, this phenomenon highlights the advantages and limitations of the numerical method. The numerical results show qualitative and sometime quantitative agreement with experiments. Furthermore, we develop general criteria for the occurrence of partial coalescence, which are very difficult to explore experimentally.