The problem concerns an elastic body containing ellipsoidal cavities with principal axes a, b, and c (a ≧ b ≫ c) which are closed by pressure and sheared. It is found that the deformed cavity remains ellipsoidal in shape, so that closure occurs simultaneously over the whole boundary, but the deformed axes are rotated with respect to the undeformed axes. The pressure required to close a cavity, in a material with Poisson's ratio v and shear modulus μ is μ(c/b)ℰ/(1 - v) (where ℰ is the elliptic integral of the second kind with modulus k2 = 1 - b2/a2 and argument π/2, and 1 ≦ ℰ ≦ π/2). A relationship is obtained between the cavity closure stress (PC33) and the uniaxial tensile strength (K*1) for a brittle body, and it is shown that PC33 ≏ 10K*1 which is in accord with experimental observations. The uniform tractions acting on the surface of the closed cavity are found to be PA13, PA23 and PA33 + PC33 where the PAij are the applied stresses (refer to the cavity axes) at large distances from the cavity.