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In the classification program of `subfactors' (i.e., inclusions of certain types of von Neumann algebras), the most important invariant is the `standard
invariant' which had been reformulated by Vaughan Jones in terms of pictures on a plane, which he named `Planar Algebras'. Since its inception, planar algebras
turned out to be a very useful tool in the theory of subfactors, namely in constructing new subfactors, detecting the existence of subfactors associated with
certain graphs, etc.
In this talk I will first describe these objects and the connection between them as given by Jones' theorem. I will then discuss a few recent results obtained in
this area.
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