Singular integral operators, which are a priori signed and non-local, can be dominated in norm, pointwise, or dually, by sparse averaging operators, which are in contrast positive and localized. The most striking consequence is that weighted norm inequalities for the singular integral follow from the corresponding, rather immediate estimates for the averaging operators. In this talk, we present several positive sparse domination results of singular integrals falling beyond the scope of classical Calderón-Zygmund theory; notably, modulation invariant multilinear singular integrals including the bilinear Hilbert transforms, variation norm Carleson operators, matrix-valued kernels, rough homogeneous singular integrals and critical Bochner-Riesz means, and singular integrals along submanifolds with curvature. Collaborators: Amalia Culiuc, Laura Cladek, Jose Manuel Conde-Alonso, Yen Do, Yumeng Ou, Yannis Parissis and Gennady Uraltsev