In this chapter, we study the existence of weak solutions of non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier-Stokes systems in a bounded Lipschitz domain in \({\mathbb R}^n\) with a transversal Lipschitz interface that intersects the boundary of the domain. For this purpose, we analyze the well-posedness of various Stokes and divergence type problems. As in the previous two chapters, we consider the relaxed ellipticity and symmetry conditions. We prove the well-posedness of the linear problems. We also prove some existence results for the nonlinear problems in \(L^2\) -based Sobolev spaces. In addition to their mathematical interest, the anisotropic Stokes and Navier-Stokes interface problems analyzed in this chapter can describe multi-phase flows of immiscible fluids with variable anisotropic viscosity tensors and compressibility influences. These problems are motivated by various industrial, biological, medical, and environmental applications.