Extension of spectroscopic rotating-analyzer ellipsometry with automated compensator function to generalized ellipsometry (GE) is reported in order to define and determine three normalized elements of the optical Jones reflection or transmission matrices r or t, respectively. These elements can be measured regardless of the specific structural and/or anisotropic properties of a nondepolarizing sample. An analytically processed 4 x 4-matrix algebra based on the Berreman 4 x 4 formalism is reviewed to calculate the Jones reflection and transmission matrix elements for arbitrarily anisotropic and homogeneously layered systems. Special solutions are available for continuously twisted biaxial media (chiral liquid crystals), and materials with arbitrary antisymmetric dielectric properties (e.g. free-carrier magneto-optics in semiconductors). The combination of both the 4 x 4-matrix algorithm and GE allows for the analysis of complex multilayered samples with inherent and arbitrarily oriented anisotropies. We present our first applications of GE to birefringent layered dielectrics (TiO2), chiral liquid crystals, spontaneously ordered semiconductor III-V compounds (AlxGa1-xInP2), and polycrystalline boron nitride thin films.