In this paper we provide fully covariant proofs of some theorems on shearfree perfect fluids. In particular, we explicitly show that any shear-free perfect fluid with the acceleration proportional to the vorticity vector (including the simpler case of vanishing acceleration) must be either non-expanding or non-rotating. We also show that these results are not necessarily true in the Newtonian case, and present an explicit comparison of shear-free dust in Newtonian and relativistic theories in order to see where and why the differences appear.1 This paper deals with shear-free perfect-fluid solutions of Einstein's field equations. The motivation for this study comes, on the one hand, from some studies on kinetic theory (see [37] and references therein), and on the other, from the relations between relativistic cosmology and Newtonian cosmology. Concerning the former, when we consider isotropic solutions of the Boltzmann equation, that is, those for which there is a timelike congruence with u as the unit tangent vector field such that the distribution function has the form f (x a , E) with E ≡ −u a p a (where p a denotes the particle momentum), two important results follow: i) The energy-stress tensor computed from such a distribution has the perfect-fluid form with respect to u (see for instance [33,35]). ii) The unit tangent vector field is shear-free and in addition its expansion θ and rotation ω satisfy ωθ = 0 (see the proof in [37]). These results led to the formulation of a conjecture whose origin seems to be the Ph. D. thesis by Treciokas [36] (see [27] for more details). This conjecture can be expressed in the following form (here ̺ and p are the energy density and pressure of the perfect fluid):Conjecture 1: In general relativity, if the velocity vector field of a barotropic perfect fluid (̺ + p = 0 and p = p(̺)) is shear-free, then either the expansion or the rotation of the fluid vanishes.While we are still probably a long way from settling the truth or falsity of Conjecture 1, it is something short of amazing that such a conjecture might be expected at all in general relativity. Consider for example the pressure-free (dust) case for which Ellis [15] showed that σ = 0 =⇒ θω = 0. This is a purely local result to which no corresponding Newtonian result appears to hold, as counterexamples can be explicitly exhibited [22]. Ellis's theorem holds for arbitrarily weak fields and fluids of arbitrarily low density. Why then does the Newtonian approximation fail?Knowing whether or not this conjecture is true, or at least to what extent it is valid, might be useful in seeking and studying new perfect-fluid solutions of Einstein's field equations with a shear-free velocity vector field. With respect to this subject, there are some interesting studies of shear-free perfect-fluid models to be found in [2,9,10,11]. On the other hand, it is important to remark that there are many known cases which are shear-free and either rotation-free or expansion-free. Some examples are: the Friedmann-Lemaître-Robertson-Walker space-...