Expressions are given for the maximum and minimum values of Poisson's ratio ν for materials with cubic symmetry. Values less than −1 occur if and only if the maximum shear modulus is associated with the cube axis and is at least 25 times the value of the minimum shear modulus. Large values of occur in directions at which the Young modulus is approximately equal to one half of its 111 value. Such directions, by their nature, are very close to 111. Application to data for cubic crystals indicates that certain Indium Thallium alloys simultaneously exhibit Poisson's ratio less than −1 and greater than +2.