Abstract: The exceptional properties of the octonion algebra allow us to define the notion of a $G_2$ structure on an oriented spin 7-manifold, which is a certain nondegenerate'' 3-form that induces a Riemannian metric in a nonlinear way. The manifold is called a $G_2$ manifold if the 3-form is parallel. Such manifolds are always Ricci-flat, and are of interest in physics. More recently, however, there has been interest in G2 conifolds'', which have a finite number of isolated cone-like'' singularities. We will begin with an introduction to $G_2$ manifolds for a general audience, paying particular attention to the similarities and differences of $G_2$ geometry with respect to the geometries of K\"ahler manifolds and of 3-manifolds. Then we will define $G_2$ conifolds, and discuss some results about them, including their desingularization and their deformation theory.