Title: Necessary and sufficient conditions in weighted theory

Abstract: Starting with the L^p boundedness of the Hilbert transform by Riesz in 1928 we go through the development of weighted theory. First Muckenhoupt and the necessary and sufficient A_p condition for one weight inequalities, then Sawyer with the testing conditions on two weight inequalities leading up to a recent two weight local Tb theorem which is a joint work with Sawyer,Shen,Uriarte-Tuero and Paparizos. I will talk about my recent work (arXiv:2009.12091) on both one and two weight inequalities providing implications on all the one and two weight necessary and sufficient conditions.