To distinguish it clearly from nonlinear viscoelasticity, we define "ideal thixotropy" as "a time-dependent viscous response to the history of the strain rate, with fading memory of that history," endowing such fluids with memory but no elasticity. An "ideal thixotropic fluid" has instantaneous stress relaxation upon cessation of flow and no elastic recoil on removal of stress. We describe "nonideal thixotropic" fluids as those whose viscoelastic time scales governing stress relaxation are much shorter than those governing the thixotropic response. This ensures that a clear distinction can be maintained between "thixotropy" and "nonlinear viscoelasticity." The stress tensor for an ideal thixotropic fluid can in general be expressed as a contraction product of a fourth rank viscosity tensor with the velocity gradient tensor, in which the viscosity tensor depends on the history of the flow. We show examples of constitutive equations that meet the definitions of ideal thixotropy or nonideal thixotropy. We also show examples of constitutive equations that have been designated as "thixotropic" by virtue of containing an equation for evolution of a "structure parameter," but whose behavior is indistinguishable from that of ordinary nonlinear viscoelasticity, and so should not be considered thixotropic. (C) 2015 The Society of Rheology.