The Boone-Higman conjecture (1973) predicts that every finitely generated group with solvable word problem embeds as a subgroup of a finitely presented simple group. There has been a flurry of activity around this conjecture in the last 5 years or so. I will survey what is known, including some joint work with Belk, Bleak, and Matucci proving the conjecture for all hyperbolic groups, and delve into some details of a new sufficient (and I would conjecture necessary) condition for a group to satisfy this conjecture.