A convex polyhedron is called monostable if it can rest in stable position only on
one of its faces. The aim of this paper is to investigate three questions of Conway,
regarding monostable polyhedra, which first appeared in a 1969 paper of Goldberg and
Guy. In this note, we answer two of these problems and make a conjecture about the
third one. The main tool of our proof is a general theorem describing approximations
of smooth convex bodies by convex polyhedra in terms of their static equilibrium points.
As another application of this theorem, we prove the existence of a convex polyhedron
with only one stable and one unstable point.