Assistive robots that address different impairments and continuously changing capabilities by providing manifold interactions based on profiles are not targeted in current research. Substantiated by a survey of human-robot interaction for assistive systems, we claim the need for establishing the innovative research topic of Accessible Human-Robot Interaction (AHRI). It is shown that limitations or loss of motoric, sensory, and mental capabilities as well as multiple impairments and resulting barriers regarding interaction, communication and perception are not sufficiently incorporated in designs of assistive robots. To achieve a design for all, fundamental research on varying needs, capabilities, and preferences along with derivation of profiles and developments of new adaptive and adaptable interaction concepts is necessary.