Skew Howe duality and categorified knot invariants I

Skew Howe duality is based on a very simple observation: the set of n by m matrices has commuting action of SL_n and SL_m.  We can use this observation to study morphisms of GL_m representations using GL_n.  This perspective has proven very useful in recent years for studying quantum knot invariants and their categorifications.  I will survey work in this direction from the last 10 years, including more recent developments concerning annular skew Howe duality and annular knot invariants.