I will first give an introduction to and brief history of G2 geometry, to compare and contrast it to Calabi-Yau geometry. G2 manifolds are important in physics because they admit parallel spinors. It is of interest to construct compact examples with singularities. I will then give a survey of some of my work that is related to conical singularities of G2 manifolds, including: desingularization, deformation theory, and a possible strategy to construct such G2 conifolds. This will include some (separate) joint works with Dominic Joyce and Jason Lotay. No previous exposure to G2 geometry will be assumed, but the focus will be more mathematical than physical. I am hoping that some of you can teach me more physics during the day.