Structure is a fundamental concept in linear algebra: matrices arising from applications often inherit a special form from the original problem, and this special form can be analysed and exploited to design efficient algorithms. In this short course we will present some examples of matrix structure and related applications. Here we are interested in data-sparse structure, that is, structure that allows us to represent an n × n matrix using only O(n) parameters. One notable example is provided by quasi separable matrices, a class of (generally dense) rank-structured matrices where off-diagonal blocks have low rank. We will give an overview of the properties of these structured classes and present a few examples of how algorithms that perform basic tasks - e.g., solving linear systems, computing eigenvalues, approximating matrix functions - can be tailored to specific structures.