Analysis, visualization, and design of vector fields on surfaces have a wide variety of applications in both scientific visualization and computer graphics. On the one hand, analysis and visualization of vector fields provides critical insights to the flow data stemming from simulation or experiments of various engineering processes. On the other hand, many graphics applications require vector fields as input to drive the following processes. This book addresses vector field analysis and design for both visualization and graphics applications. More specifically, two topology representations of 2D vector fields, ECG and MCG, are introduced to address the completeness and accuracy issues in vector field analysis. In order to assist many modern graphics applications involving dynamical systems, a design system for time-varying vector fields on surfaces is presented for the first time. In summary, the presented work not only enriches the knowledge of vector field related study but also provides a number of future directions for this community.