We describe a new way to decompose one-electron orbitals of a molecule into atom-centered or fragment-centered orbitals by an approach that we call "maximal orbital analysis" (MOA). The MOA analysis is based on the corresponding orbital transformation (COT) that has the unique mathematical property of maximizing any sub-trace of the overlap matrix, in Hilbert metric sense, between two sets of nonorthogonal orbitals. Here, one set comprises the molecule orbitals (Hartree-Fock, Kohn-Sham, complete-active-space, or any set of orthonormal molecular orbitals), the other set comprises the basis functions associated with an atom or a group of atoms. We show in prototypical molecular systems such as a water dimer, metal carbonyl complexes, and a mixed-valent transition metal complex, that the MOA orbitals capture very well key aspects of wavefunctions and the ensuing chemical concepts that govern electronic interactions in molecules. © 2019 Wiley Periodicals, Inc.