This presentation will give a short tour through the Canterbury Zoo of Spaces, i.e. it will introduce the basics and some nice examples and applications of Differential Geometry and its subfields Riemannian Geometry, Complex Geometry and Symplectic Geometry. Differential Geometry considers manifolds, which are spaces that are locally approximated by R^n, just as we can consider the Earth flat as long as we work on small scales. The three subfields mentioned all consider manifolds endowed with an additional structure that give them interesting properties and allow for special analysis techniques. In the course of the tour we will introduce the main tools of differential geometry and its subfields such as vector fields, differential forms, Riemannian metrics, complex structures and symplectic forms.

Disclaimer: This presentation is intended for the widest possible audience, with as many motivating examples and images as possible.