One of the major open problems in nonhomoheneous Harmonic analysis was to find the necessary and sufficient conditions for the boundedness of two weights inequality for singular integrals. In this talk, we will first discuss the famous two weights problem for the Hilbert transform and outline our proof that verifies this longstanding open problems (40 years). We then discuss some of the main difficulties for higher dimensional singular integrals. Some of the important applications to other fields will be also discussed, in particular an application of our T1 theorem for the Cauchy transform in the complex plane settled the longstanding open problem; the embedding problem for the model spaces.