The notion of `planar algebras' is due to Vaughan Jones, who also showed (using earlier work of Sorin Popa) that planar algebras are equivalent to (and constitute a reformulation with a distinctly topological flavour of) the so-called `standard invariant of a subfactor of finite index' (which has a distinctly analytical flavour). The aim of the talk is to convey some of this topological flavour, and to illustrate the notion with examples of some planar algebras and of some of their sub-planar algebras.