The theory of the elastic fields round ellipsoidal inclusions and inhomogeneities together with the well-known analogy between linear elasticity and slow incompressible viscous flow are used to develop the governing equations for the finite deformation of a viscous ellipsoidal inhomogeneity in a viscous matrix undergoing a general linear time-dependent flow at infinity. The governing equations are then solved for an inhomogeneity in the form of an elliptic cylinder in a linear two-dimensional flow whose stream lines at infinity are steady. The behaviour of the inhomogeneity under pure shear and simple shear is considered in detail and it is shown that the boundaries of certain deforming inhomogeneities remain unchanged during simple shear. These steady inhomogeneities can appear also in general linear two-dimensional applied flows. In such flows the behaviour is influenced both by the initial shape and orientation of the inhomogeneity and by its viscosity. Inhomogeneities which are rather viscous or subject to an applied flow with high vorticity deform periodically, while most others elongate indefinitely. The patterns of behaviour may be described in terms of a number of regimes which can be classified by considering the singularities of the differential equations governing the variations of shape and orientation of the inhomogeneity, or, equivalently, by studying the invariants of the corresponding one-parameter Lie groups. Finally, some obvious extensions of the treatment are indicated. These make it possible to consider inhomogeneities (such as holes) whose volume does not remain constant, and which have constitutive relations more general than those of a linear viscous material.