Haohua Deng of Dartmouth College presents recent work on constructing completions of period mappings in Hodge theory, delivered at the Institute for Advanced Study's workshop on Hodge theory and o-minimality. He reviews the well developed theory for the classical case, where the period domain is Hermitian symmetric, before turning to non-classical settings where that structure breaks down. The core of the talk covers generalized toroidal completions, following the framework of Ash, Mumford, Rapoport and Tai and later work of Kato, Nakayama and Usui, and presents new results from joint work with Colleen Robles and Jacob Tsimerman. The talk is aimed at specialists in algebraic and differential geometry already familiar with period domains, variations of Hodge structure, and the technical machinery needed to extend these mappings to their boundaries. It is a research seminar talk rather than an introductory survey, assuming graduate-level background in Hodge theory.