This paper is an expanded version of a talk given at the Current Developments in Mathematics Conference last November (2002) on the work of Wilfred Schmid on periods of limits of Hodge structures. The paper begins with an exposition of the theory of limits of Hodge structures with some emphasis on the geometry. It goes on to discuss (briefly) the periods of the limit mixed Hodge structure on the fundamental group (made unipotent) of the projective line minus 3 points and its connections with mixed zeta numbers and mixed Tate motives over Spec Z through the work Deligne, Goncharov and others.

27 pages, 1 figure, to appear in the proceedings of the CDM Conference