Since they were introduced about two decades ago, symplectic singularities have shown themselves to be a remarkable branch of algebraic geometry.  They are much nicer in many ways than arbitrary singularities, but still have a lot of interesting nooks and crannies.

I'll talk about these varieties from a representation theorist's perspective.  This might sound like a strange direction, but remember, any interesting symplectic structure is likely to be the classical limit of an equally interesting non-commutative structure, whose representation theory we can study. While this field is still in its infancy, it includes a lot of well-known examples like universal enveloping algebras and Cherednik algebras, and has led a lot of interesting places, including to categorified knot invariants and a conjectured duality between pairs of symplectic singularities.  I'll give a taste of these results and try to indicate some interesting future directions.