In this chapter we define the general notion of a fibre bundle. This is, roughly speaking, a space that locally looks like a product. Whilst fibre bundles are important in many areas of topology, they are slightly too vague to be useful for us. We therefore quickly specialise to the two special types of fibre bundles used in differential geometry: vector bundles and principal bundles. The study of such bundles will make up the majority of the rest of Volume I.