In this thesis, we will develop an implementation of Olsson and Kreiss’s conservative level set method. Do achieve this, we must cover key aspects of fluid dynamics, numerical methods for PDEs, including the finite element method. Prior to discussing aspects of two-phase flow, we first overview the foundational principles of fluid dynamics, as detailed in chapter 2. This chapter entails derivations of the governing equations of fluid flow which follow from conservation of mass and momentum, as well as some fundamental solutions to them. In chapter 3, we discuss how approximate solutions the Naiver-Stokes equation may be obtained via a splitting integration method and a projection step. To complete this, we discuss how the Finite Element Method is used to find the numerical solution a general boundary value problem, allowing us to solve the equations arising from projection methods for the Naiver-Stokes equations. This will give us the necessary tools for simulating two-phase flows in chapter 4, where we will see how the level set method can be used to solve the free-surface problem inherent in multiphase flows. This will leave us with a numerical method capable of tackling various problems with practical applications, some of which are detailed in chapter 5. In chapter 6, we examine one aspect of non-Newtonian fluid behavior within the context of viscoplastic or yield stress fluids. Here, we investigate the fundamentals of viscoplastic dynamics through the idealized Bingham fluids, which provide a simple model of viscoplasticity. We will also see how regularizing viscosity can be used to find the numerical solution to viscoplastic flows, allowing the modeling two-phase viscoplastic flows with a conservative level set method.