The talk will be devoted to the billiard illumination problem inside a polygon; the exact formulation will be given at the talk. This is the first problem on polygonal billiards the answer to which depends on the convexity of the polygon: if the polygon is _non-convex_, the answer is negative, even under the strongest possible hypotheses (a result of G. W. Tokarsky, 1995).

We conjectured (first in 1994 and again at the Urbana AMS meeting in 1999) that the answer to the question is always positive for convex polygons, even under the weakest possible hypotheses, and we proved the conjecture for squares.

We will explain the difference between the two cases and present our proof of the illumination problem in the square and its generalization for the geodesic flow on the surface of a cube.