1) A stress tensor is defined for vorticity fields in hydrodynamics, analogous to Maxwell stress tensor in electromagnetism. This vorticity stress tensor represents forces on surfaces in a vorticity field. 2) The vorticity stress tensor can be decomposed into terms representing tension along vortex lines and pressure normal to vortex lines. 3) This vorticity stress tensor establishes a theoretical framework for studying forces between vortex lines and deformation of fluid volumes due to vorticity.