Previous investigations have been concerned with the stress concentrations around internal cavities in the shape of an ellipsoid of revolution. This paper contains an exact closed solution, in terms of Jacobian elliptic functions, for the stress distribution around a general triaxial ellipsoidal cavity in an infinite elastic body. The body at infinity is in a uniform state of stress whose principal directions are parallel to the axes of the cavity, the magnitudes of the principal stresses at infinity being arbitrary. The solution covers as limiting cases the known results for spherical and spheroidal cavities. The technically important aspects of the ensuing stress concentration are discussed in detail.