We study the problem of acute triangulations of convex polyhedra and the
space R^n. Here an acute triangulation is a triangulation into simplices whose
dihedral angles are acute. We prove that acute triangulations of the n-cube do
not exist for n>=4. Further, we prove that acute triangulations of the space
R^n do not exist for n>= 5. In the opposite direction, in R^3, we present a
construction of an acute triangulation of the cube, the regular octahedron and
a non-trivial acute triangulation of the regular tetrahedron. We also prove
nonexistence of an acute triangulation of R^4 if all dihedral angles are
bounded away from pi/2.