An analytical model is presented for the deformation of an ellipsoidal Newtonian droplet, suspended in another Newtonian fluid with different viscosity and zero interfacial tension. The theory is exact for any linear velocity field, and is not limited to small deformations. It encompasses some well-known special cases, such as Jeffery's equation for solid axisymmetric particles and Taylor's small-deformation theory for droplets. Example calculations exhibit droplet stretching, reorientation, and tumbling, and provide a reasonable match to available experimental data on transient and steady droplet shapes. The corresponding rheological theory for dilute dispersions is also derived, in a form that explicitly includes the effects of microstructure on dispersion rheology.