This is an introductory lecture to the index of differential operators. We´ll discuss the analytic and topological information it carries and the theorem of Atiyah and Singer, explaining how it subsumes previous geometric results, such as the Gauss-Bonnet theorem. We´ll also describe the topics that this series of lectures is going to cover. ---- Note: This is the introduction of a course on Index theory by Prof. Androulidakis (FCUP)