This study aimed to identify and describe the fundamental characteristics of spoken dialogue systems, and their role in supporting human-robot interaction and enabling the communication between socially assistive robots and patients with dementia. First, this work provides an overview of spoken dialogue systems by considering the underlying technologies, approaches, methods, and general issues. Then, the analysis focuses on studies, systems, and approaches that have investigated the role of dialogue systems and conversational agents in the interaction with elderly people with dementia by presenting the results of a literature review. While the overview of spoken dialogue systems relies on existing surveys and reviews, a research was conducted to identify existing works in the literature that have investigated the role of conversational agents and dialogue systems in the elderly and people with cognitive impairments. Inclusion criteria were as follows: (1) use of conversational agents, dialogue systems, or language processing tools for people with cognitive impairments; (2) age ≥60 years; (3) diagnosis of dementia according to National Institute on Aging-Alzheimer's Association (NIAAA) criteria; (4) presence of tests or experiments with qualitative or quantitative results. Initially 125 studies published between 2000 and 2017 were identified, of which 12 met the inclusion criteria. The review identifies the issues and challenges that are reported when conversational agents and speech-based interfaces have been used for interacting with people with cognitive impairments. In addition, the review led to the identification of studies that have investigated speech processing and natural language processing capabilities to assess the cognitive status of people with dementia.