John Conway's Holyhedron Problem has been solved by Jade Vinson in “On Holyhedra” (with an introduction by John Conway), Discrete and Computational Geometry, volume 24, number 1, pages 85-104, 2000. There is still work to be done in order to win the $10,000 / (number of faces) reward, because Vinson's polyhedron has 78,585,627 faces and genus 60,380,421! Don Hatch reports some further progress: a 492-face holyhedron.
John Conway's Angel and Devil Problem has been solved, even for power-2 angel, by András Máthé and Oddvar Kloster. See this blog post. There are other solutions, for higher-power angels, by Brian Bowditch and Peter Gács.