Planar algebras were first introduced in the late 90's by Vaughan Jones as an axiomatization of the standard invariant of a subfactor. Jones' idea was that the structure of standard invariants had a description in terms of planar diagrams, and that one could compute things about the subfactor by manipulating the pictures. Since then, planar algebras have been used extensively as a framework for performing rigorous calculations by manipulating diagrams. In this talk I will give an example-driven introduction to planar algebras and diagrammatic calculation, and demonstrate some of the features of working with pictures. If time permits, I will also discuss the state of the on-going project to classify `small' planar algebras, as well as the role played by planar algebras in constructive quantum field theory.