A holomorphic function is, in many ways, a generalization of a polynomial.
A non-commutative holomorphic function is, in a similar way, a generalization of a non-commutative polynomial, that is a polynomial like $2 x^2 y - 3 xyx$. They occur naturally when one wishes to do functional calculus with matrices. In this talk, I will explain their history, starting with J.L. Taylor in the 1970's, why they have suddenly taken off in the last decade, and how they are connected to other areas of mathematics.