We present a complete theory of polarization measurements of second-order nonlinear light scattering in isotropic solutions of nonlinear molecules. The nonlinear interaction between the molecules and input beams at frequencies omega(1) and omega(2) gives rise to incoherently scattered radiation at frequency omega(1) + omega(2). The fundamental quantity measured by such experiments in the laboratory frame of reference is the quadratic orientational average [beta(ikl)/beta*(jmn)] of the hyperpolarizability tenser. The number of independent orientational averages that can be measured is shown to be equal to the number of rotational invariants of sixth rank quantities. The absolute maximum number of independent measurements is 15 and occurs for the most general case in which the hyperpolarizability tensor is complex and the dispersion between all three frequencies is important. This number is reduced to eleven for the case of a real hyperpolarizability tensor and to six and five for the case of hyper-Rayleigh scattering and complex and real tensors, respectively. For the case of planar molecules, these numbers are further reduced to ten, seven, five, and four, respectively. We present explicit expressions that relate the rotational invariants to the components of the hyperpolarizability tensor in the molecular frame of reference. We also present practical measurement schemes that can be used to determine all rotational invariants experimentally and discuss the possibilities and limitations of nonlinear light scattering in determining the values of individual components of the molecular hyperpolarizability tenser.