Over the last two decades, animal social network analysis has become central in the study of animal social systems. This methodology has given researchers a powerful set of tools to ask deep questions about the social structures of animals, and how these are linked to many other important biological processes. Animal social networks are often constructed from noisy, uncertain data, which would be well-suited to a Bayesian statistical philosophy. However, despite recent advances in Bayesian methodologies, they remain underutilised in animal social network analysis. In part this is due to unique features of animal network data that have led to the development and use of non-standard statistical procedures in the field. In this thesis I study some of the issues around existing methods, and highlight how a Bayesian methodology could substantially improve animal social network analyses. I introduce, implement, and explore a Bayesian framework for animal social network analysis. The framework makes it possible to conduct new types of analyses while accounting for both uncertainty and sampling biases. In addition to this, I have developed an R software package to allow researchers to use the new Bayesian framework to conduct animal social network analyses. The development of this framework raises new questions and opens up new opportunities in animal social network analysis, which I briefly explore towards the end of this thesis. I hope the developments made in this thesis will help to guide the future of animal social network analyses to make the most of hard-won network data, and to generate more reliable and insightful scientific inferences.