A unified description of topological singularities in fields is introduced to describe both crystalline solids and nematics. A phase field description of solids, and a tensor order parameter description of nematics are used to introduce topological densities and exact kinematic laws for the motion of dislocation and disclination lines. In the case of a solid, coarse grained fields become singular at defect lines, and their kinematics is described in terms of the motion of the field singularities. For a nematic liquid crystal, the tensor order parameter remains regular. Following earlier experimental and theoretical work, the disclination core is defined to be the line where the uniaxial and biaxial order parameters are equal, or equivalently, where the two largest eigenvalues of the tensor order parameter cross. This allows an exact expression relating the velocity of the line to spatial and temporal derivatives of the order parameter, which can be specified by a dynamic model. Analytical results are given for several prototypical configurations, including line interactions and motion, loop annihilation, and the response to external fields and shear flows. Behavior that follows from topological constraints or defect geometry is highlighted.