Mathematics is a crucial component of bioinformatics, playing a key role in analyzing, modelling and interpreting complex biological data. As the field of high throughput sequencing, systems biology and computational genomics has surged in development, mathematical models have emerged as a key component in moving from raw biological data to actionable knowledge. Providing a thorough introduction to the mathematics that underlie modern bioinformatics, this paper covers topics such as statistical modelling, linear algebra, graph theory, optimization and machine learning. The important applications include sequence alignment, phylogenetic reconstruction, analysis of gene expression, protein structure prediction and network biology. New mathematical methods are also mentioned, such as deep learning, stochastic approaches and topological data analysis. Finally, open challenges and future research directions in mathematics and biological data science are identified.