The past 30 years has seen an increased willingness for scientists to assess the subjective experiences of animals. There are good reasons to believe that domesticated animals are sentient beings, capable of having feelings. The study of feelings is central to the assessment of animal welfare. Feelings are not directly observable, but have measurable correlates or consequences. They can be assessed either by giving animals some control over their environments and observing the choices and decisions they make, using preference and motivation tests (including aversion tests), or by looking for signs of deprivation, frustration or distress when the animal is confined in an environment or subjected to a treatment without any means of control. Preference and motivation tests assess motivation, defined operationally as the tendency for an animal to perform a behaviour, but understood as reflecting the animal's desire to do so. These tests are used to address four distinct research questions: (1) whether an animal is motivated to obtain or avoid a resource; (2) whether it has a preference amongst alternative resources (i.e., whether it is more motivated for one than the other); (3) how strong its motivation or preference is and (4) whether its preference, or the strength of its motivation or preference, is altered by changes in its internal or external environment. Methods that have been used to answer these questions fall into two overlapping categories: choice tests and operant tests. A wide array of such tests has been employed, each with its own strengths and weaknesses. We describe individual methods, indicating the research questions they have been designed to address, list their limitations and discuss methodological problems that are not yet fully resolved. Preference and motivation tests are techniques that are still under development. However, if carefully interpreted and properly integrated with other measures of welfare, they constitute a powerful tool for the assessment of welfare.