We study the variety of common tangents for up to four quadric surfaces in projective three-space, with particular regard to congurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in the projective space dened by all complex quadric surfaces which express the fact that several quadrics are tangent along a curve to one and the same quadric of rank at least three, and called, for intuitive reasons: a basket. Lines in any ruling of the latter will be common tangents. These considerations are then restricted to spheres in Euclidean threespace, and result in a complete answer to the question over the reals: "When do four spheres allow infinitely many common tangents?".