Abstract The self-organization of monolayer epitaxy islands in presence of anisotropies in surface stress, applied stress, and lattice mismatch between the film and substrate materials is investigated. The fundamental nature of island interactions is addressed in the context of a model wherein the system free energy consists of the excess energy and strain energy of atomic surface steps. It is shown the anisotropy can change the character of island interactions. An energy-reducing kinetic relation is adopted to evolve an initially random morphology towards a generally metastable minimum energy state. It is found the self-organization of islands into a regular array requires both the repulsion between islands and tendency for islands to aligned in a particular direction. Small anisotropies provide the required repulsion but not the tendency for islands to align and large anisotropies provide the necessary alignment but cause islands to attract. Modest levels of anisotropy provide the most favorable conditions of self-organization.