The recent idea of extracting effective atomic orbitals from molecular wave functions by performing independent localization transformations for each atom separately is generalized to the case of an arbitrary Hermitian bilinear localization functional. The general equations are derived and the orthogonality relationships pertinent to the localized molecular orbitals are proved, The "intraatomic components" of these localized orbitals form an effective atomic basis, which is also automatically orthogonal if some conditions are fulfilled. Several different localization functionals are considered and it is shown that for the simplest one the orbitals obtained are natural hybrids in McWeeny’s sense and are conceptually close to (but not identical with) Weinhold’s "natural hybrid orbitals". In this case one obtains for each atom of a "usual" molecule as many effective AOs of appreciable importance as the number of orbitals contained in the classical "minimal basis" of that atom, forming therefore a-distorted but still orthogonal-effective minimal basis of the atom within the molecule. The similarities and differences with Weinhold’s "atomic natural orbitals" are also discussed. It is pointed out that by selecting a proper localization functional, the present approach can also be used to define the effective atomic orbitals in a basis-free manner, i.e. even if no atom-centered basis was used in calculating the wave function. A possibility of generalization for correlated wave functions is also suggested.