In this paper, we present a fractional dashpot that is capable of continuously adjusting the degree of dissipation between the whole dashpot and the Maxwell model (spring and dashpot in serial connection). It is related to the springpot by a frequency shift of the relaxation modulus in the complex domain. In contrast to the springpot, which is a viscoelastic solid-like model, the fractional dashpot is an intrinsically viscoelastic fluid model. We demonstrate that the fractional dashpot allows one to better model the polydispersity effect of typical viscoelastic fluids, overcoming undesired stationary predictions and reducing or even eliminating multiple modes in data fitting. The linear version of the fractional dashpot model is also scaled up to large deformation by incorporation of corotational tensor derivatives and a projected strain rate tensor for constructing an objective fractional constitutive equation. The resulting model having five model parameters (one for fractionality, two for linear viscoelasticity, one for straining, and the last one for rotation) is able to fit startup shear viscosity of a high molecular weight polystyrene solution in high accuracy, and yet using only a single mode. (C) 2018 The Society of Rheology.