I will talk about a special class of closed G2-structures, those satisfying the `quadratic' condition. This is a second order PDE system first written down by Bryant that can be interpreted as a condition on the Ricci curvature of the induced metric, that includes the extremally Ricci-pinched (ERP) condition as a special case. I will talk about various constructions of quadratic closed G2-structures, including the first examples of ERP closed G2-structures that are not locally homogeneous and the first examples of quadratic closed G2-structures that are not ERP. I will discuss the relationship with the Laplace flow, and give new examples of Laplace solitons.