One of the main objectives of robotic communities is developing platforms that could be included in our every day life as helping tools. This aim is especially pursued in healthcare, with a particular focus on elderly care. With the older population growing exponentially, the challenges that healthcare systems are facing to provide good quality care to older adults are straining the systems to the point that, sometimes, the well-being of older adults could be jeopardized. For this reason, socially assistive robots are starting to be deployed and evaluated as a means to complement the work of carers. Robots could be efficient tools to help taking care of the older population through social interactions, with the hope to fight isolation and loneliness which are the main causes for physical and cognitive decline. The aim of the thesis is to explore possible ways in which socially assistive robots could be used to provide companionship to older adults. To do so, different approaches to companionship have been investigated. However, to arrive to the development of effective socially assitive robots and applications, it is important, as a first step, to understand the needs and preferences of the population for whom these platforms are developed. Therefore, the first step of this research was to initiate a conversation with the prospective users of socially assistive robots to better understand what they expect from robots and how they would like to interact with them. This user-centred approach was followed throughout the experimental phases of the research described in this thesis: every system was developed and tested following older adults' feedback. The first approach to companionship that was investigated in this thesis was a more indirect one, aiming to use human-robot interactions to transparently monitor the physical and cognitive well-being of older adults, with the objective to facilitate early diagnosis of frailty or mild cognitive impairment. Following this approach, it was examined whether a socially assistive robot would be accepted by older adults living alone as a monitoring tool always vigilant in their own houses. To do so, a study deploying a socially assistive robot in the house of older adults was run for three months. The second, more direct approach to companionship, aimed to improve the social intelligence of robots by letting them learn directly from their interactions with the users. In particular, by equipping a social assistive robot with the ability to autonomously decide whether to interact with their users and in what way (using deep learning architectures to analyse nonverbal bodily social signals). This approach was tested with a human-robot interaction experiment aiming to make the robot understand when was it acceptable to start a conversation with its users, and by a comparative study on different deep learning architectures to find the one that could most successfully predict the apparent personality traits of the robot's users, to provide a baseline of behaviour for the robot to adapt at the initiation of contact with its users. The results gathered through the work done in this thesis highlight how important and valuable the research efforts towards the inclusion of socially assistive robots in the life of older adults could be. The systems developed and described throughout this thesis show promise in what they were able to deliver, and they were well received by the older adults who participated in the human-robot interactions experiments. The most important take away message that can be derived from this work is that older adults are more than willing to use robots and assistive technologies. Moreover, using deep learning to autonomously facilitate interactions by making robots learn from nonverbal bodily social signals is a novel approach that revealed itself to have potential, since the deep learning agent was able to improve its performance progressively and showed its ability to learn some apparent personality traits. Therefore, these results show how, through the use of socially assistive robots, it could be possible to monitor the physical and cognitive well-being of older adults, through interactions aiming to assess early decline, and how an additional step towards more personalised interaction could be taken using deep learning.