The goal of this thesis is to explore the basic axiomatic theory of Syn- thetic Differential Geometry (SDG). This field aims to put the study of smooth manifolds, and geometry therein, in a topos-theoretic framework. Though the full depth of application and consequences of SDG require knowledge of topos theory to comprehend, a large part of the theory can be appreciated with only some notions of basic category theory (as well as with a standard undergraduate mathematics syllabus). In this work we look at this part of SDG, called the axiomatic the- ory because it is indeed developed axiomatically. Specifically, under the axiomatic theory of SDG we look at differential calculus, then manifolds (their analogue in SDG), vector bundles (the tangent bundle as a particular case), and vector fields (and Lie algebras thereof).