What are skew monoidal categories?

Abstract: 

Many structures in mathematics admit some notion of tensor product.  For example, we have the cartesian product of sets, the tensor product of vector spaces and of chain complexes.  The common abstract features of such tensor products are captured by the useful concept of a monoidal (aka tensor) category.

In 2012 a rather curious generalisation known as a skew monoidal category was introduced by Szlachanyi.  I will introduce these skew monoidal categories, describe some recent research on them and discuss some of their future prospects as time permits.