V.I.ARNOLD HIGHER DIMENSIONAL CONTINUED FRACTIONS September 2, 1998 (REGULAR AND CHAOTIC DYNAMICS, V.3, No 3, 1998) The higher-dimensional analogue of a continuous fraction is the polyhedral surface, bounding the convex hull of the semigroup of the integer points in a simplicial cone of the euclidian space. The article describes some conjectures and theorems, extending to such higher-dimensional continouos fraction the Lagrange theorem on quadraticirrationals and the Gauss--Kuzmin statistics.