Use of accelerometers is now widespread within animal biologging as they provide a means of measuring an animal's activity in a meaningful and quantitative way where direct observation is not possible. In sequential acceleration data, there is a natural dependence between observations of behaviour, a fact that has been largely ignored in most analyses. Analyses of acceleration data where serial dependence has been explicitly modelled have largely relied on hidden Markov models (HMMs). Depending on the aim of an analysis, an HMM can be used for state prediction or to make inferences about drivers of behaviour. For state prediction, a supervised learning approach can be applied. That is, an HMM is trained to classify unlabelled acceleration data into a finite set of pre-specified categories. An unsupervised learning approach can be used to infer new aspects of animal behaviour when biologically meaningful response variables are used, with the caveat that the states may not map to specific behaviours. We provide the details necessary to implement and assess an HMM in both the supervised and unsupervised learning context and discuss the data requirements of each case. We outline two applications to marine and aerial systems (shark and eagle) taking the unsupervised learning approach, which is more readily applicable to animal activity measured in the field. HMMs were used to infer the effects of temporal, atmospheric and tidal inputs on animal behaviour. Animal accelerometer data allow ecologists to identify important correlates and drivers of animal activity (and hence behaviour). The HMM framework is well suited to deal with the main features commonly observed in accelerometer data and can easily be extended to suit a wide range of types of animal activity data. The ability to combine direct observations of animal activity with statistical models, which account for the features of accelerometer data, offers a new way to quantify animal behaviour and energetic expenditure and to deepen our insights into individual behaviour as a constituent of populations and ecosystems.