The authors study the topology of symmetric, second-order tensor fields. The results of the study can be readily extended to include general tensor fields through linear combination of symmetric tensor fields and vector fields. The goal is to represent their complex structure by a simple set of carefully chosen points, lines, and surfaces analogous to approaches in vector field topology. They extract topological skeletons of the eigenvector fields and use them for a compact, comprehensive description of the tensor field. Their approach is based on the premise: "analyze, then visualize". The basic constituents of tensor topology are the degenerate points, or points where eigenvalues are equal to each other. Degenerate points play a similar role as critical points in vector fields. In tensor fields they identify two kinds of elementary degenerate points, which they call wedge points and trisector points. They can combine to form more familiar singularities-such as saddles, nodes, centers, or foci. However, these are generally unstable structures in tensor fields. Based on the notions developed for 2D tensor fields, they extend the theory to include 3D degenerate points. Examples are given on the use of tensor field topology for the interpretation of physical systems.