This paper reviews issues and developments in the field of structure-environment interaction problems, in which the environment is an elastic body, possibly unbounded. It covers in particular the fields of soil-structure interaction, ground-borne noise and vibration emitted by transportation systems and wave diffraction by obstacles in an elastic medium. The general setting for a linear bounded structure coupled to an unbounded linear environment through a bounded interface is first recalled, and the domain decomposition technique classically used for its description is put up. Extensions for the cases of non-linear structure and uncertain environment are then discussed. Finally, the ongoing research in the fields of unbounded interface, moving interface, and multiple interfaces are reviewed and summarized.