This paper presents a survey of the state-of-the-art in motion estimation of continuum objects such as elastic solids (e.g., human organs, skin, and biomedical tissues) and fluids (e.g., clouds, ocean flows, and air flows). The survey is focused on the estimation methodologies, rather than on specific applications. We begin by summarizing the main components and challenges of continuum-motion estimation, and structure our review around these components. We provide a classification of the related work according to their solutions to these challenges. A discussion on the methodologies for quantitative evaluation of methods is also provided. Finally, we conclude the survey by pointing out some open problems in continuum-motion estimation.