As datasets grow ever larger in scale, complexity and variety, there is an increasing need for powerful machine learning and statistical techniques that are capable of learning from such data. Bayesian nonparametrics is a promising approach to data analysis that is increasingly popular in machine learning and statistics. Bayesian nonparametric models are highly flexible models with infinite-dimensional parameter spaces that can be used to directly parameterise and learn about functions, densities, conditional distributions etc, and have been successfully applied to regression, survival analysis, language modelling, time series analysis, and visual scene analysis among others. However, to successfully use Bayesian nonparametric models to analyse the high-dimensional and structured datasets now commonly encountered in the age of Big Data, we will have to overcome a number of challenges. Namely, we need to develop Bayesian nonparametric models that can learn rich representations from structured data, and we need computational methodologies that can scale effectively to the large and complex models of the future. We will ground our developments in relevant applications, particularly to natural language processing (learning distributed representations for language modelling and compositional semantics) and genetics (modelling genetic variations arising from population, genealogical and spatial structures).