Directional statistics has progressed significantly since the early 1950s; see e.g. Mardia and Jupp (2000), Jammalamadaka and SenGupta (2001) and Pewsey et al. (2013). The present book presents selected results from the last two decades, which respond to needs from areas such as bioinformatics, machine learning, cosmology and others. Like Pewsey et al. (2013), it builds upon Mardia and Jupp (2000) and Jammalamadaka and SenGupta (2001). However, while Pewsey et al. (2013) focuses on circular data and R with practical applications, this book emphasises theoretical developments, with a more applied companion book Applied Directional Statistics: Modern Methods and Case Studies to follow.