Contact geometry is the study of certain geometric structures on odd dimensional differentiable manifolds, which originated in classical mechanics.  Manifolds generalize the notion of curves (1 dimensional) and surfaces (2 dimensional) to higher dimensions, though for this talk we will stick to dimension three.  A contact structure is a two dimensional subspace which satisfies a maximum non-degeneracy condition which mandates that there be no two dimensional subsurface which is tangent to the contact structure.   It also gives rise to a canonically defined vector field, called the Reeb vector field.  I will give an introduction to three dimensional differential and contact geometry, primarily focusing on R^3 and S^3 as examples, with numerous pictures and animations.