Compact torsion-free $G_2$-manifolds are 7-dimensional analogues of Calabi-Yau threefolds, being compact Ricci-flat Riemannian manifolds with reduced holonomy that are important ingredients in theories of physics. All known constructions use an abstract existence theorem of Dominic Joyce to perturb "almost" solutions of a quasilinear elliptic PDE to honest solutions, and construct the "almost" solutions via glueing methods. I will first summarize some basic facts about $G_2$-manifolds and Joyce's existence theorem, and then briefly mention the previous constructions by Joyce (1994), Kovalev (2003), and Corti-Haskins-Nordstrom-Pacini (2014). Then I will focus on a new construction (joint work of myself and Joyce, 2017) that is significantly more involved for several reasons, which I will elucidate. In particular one key step in our construction involves solving a linear first order elliptic PDE on a noncompact 4-manifold with prescribed asymptotics at infinity. (arXiv: 1707.09325)