Topology-related issues are becoming increasingly important in Computer Graphics. This research examines the use of topological analysis for solving two important problems in 3D Graphics: surface parameterization, and vector field design on surfaces. Many applications, such as high-quality and interactive image synthesis, benefit from the solutions to these problems. Surface parameterization refers to segmenting a 3D surface into a number of patches and unfolding them onto a plane. A surface parameterization allows surface properties to be sampled and stored in a texture map for high-quality and interactive display. One of the most important quality measurements for surface parameterization is stretch, which causes an uneven sampling rate across the surface and needs to be avoided whenever possible. In this thesis, I present an automatic parameterization technique that segments the surface according to the handles and large protrusions in the surface. This results in a small number of large patches that can be unfolded with relatively little stretch. To locate the handles and large protrusions, I make use of topological analysis of a distance-based function on the surface. Vector field design refers to creating continuous vector fields on 3D surfaces with control over vector field topology, such as the number and location of the singularities. Many graphics applications make use of an input vector field. The singularities in the input vector field often cause visual artifacts for these applications, such as texture synthesis and non-photorealistic rendering. In this thesis, I describe a vector field design system for both planar domains and 3D mesh surfaces. The system provides topological editing operations that allow the user to control the number and location of the singularities in the vector field. For the system to work for 3D meshes surface, I present a novel piecewise interpolating scheme that produces a continuous vector field based on the vector values defined at the vertices of the mesh. I demonstrate the effectiveness of the system through several graphics applications: painterly rendering of still images, pencil-sketches of surfaces, and texture synthesis.