Vector fields naturally arise in numerous areas of science and engineering, such as fluid dynamics, aerodynamics, electromagnetism, computer vision and biology. A popular approach to vector field analysis is vector field topology, whose goal has traditionally been to describe vector field data in terms of features such as stationary points, periodic orbits and separatrices. Designing robust algorithms for finding such features and estimating their importance is one of the important fundamental problems in scientific visualization. Classical approaches to this problem are built upon the foundation provided by classical dynamical systems theory. Roughly speaking, numerical methods are used to approximate trajectories of the vector field and features are a result of analysis of the structure of these approximations. While these approaches have been used to obtain beautiful visualizations, they have been hindered by high complexity of the output, sensitivity to numerical error, high computational cost and consistency issues. This talk will provide an overview of an alternative, purely computational geometric framework based on discontinuous (piecewise constant) vector fields. Our approach guarantees consistency of the output, can be used to estimate stability of features with respect to perturbation of the input and supports multi-scale analysis of vector field topology.