It has been observed in recent years that many singular varieties admit a resolution of singularities, in a suitable sense, given by a non-commutative sheaf of algebras. This sometimes happens in cases where no nice commutative resolution is known. Even if a good commutative resolution is known to exist the non-commutative resolution may still be smaller. In the talk I will give a general introduction to this phenomenon. In particular I will discuss some recent joint work with Spela Spenko about non-commutative resolutions of determinantal varieties for symmetric and skew-symmetric matrices