When studying deformations of singular varieties, a fundamental question is whether they admit smoothings. In this talk, I will describe new results in this direction for higher-dimensional singular Fano and Calabi-Yau varieties with Du Bois isolated complete intersection singularities. We obtain smoothing criteria by imposing local conditions on the singularities, expressed in terms of the recently introduced notions of higher rational and higher Du Bois singularities. In the Calabi-Yau case, one must also impose a global condition, which we formulate in terms of the Hodge-Du Bois numbers of the variety. Our results recover and extend various theorems of Friedman, Namikawa, Namikawa-Steenbrink, Gross, and Friedman-Laza.