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.