  In this article we give an overview of the problem of finding sharp constants
in matrix weighted norm inequalities for singular integrals, the so-called
matrix A2 conjecture. We begin by reviewing the history of the problem in the
scalar case, including a sketch of the proof of the scalar A2 conjecture. We
then discuss the original, qualitative results for singular integrals with
matrix weights and the best known quantitative estimates. We give an overview
of new results by the author and Bownik, who developed a theory of harmonic
analysis on convex set-valued functions. This led to the proof the Jones
factorization theorem and the Rubio de Francia extrapolation theorem for matrix
weights, two longstanding problems. Rubio de Francia extrapolation was expected
to be a major tool in the proof of the matrix A2 conjecture; however, this
conjecture was very recently proved false. We discuss this problem.
