The subject of this talk is the double shuffle Lie algebra associated to multizeta values. If one multiplies two multizeta values, one obtains a sum of multizeta values, with coefficients in Q, of multizeta according to two distinct multiplication laws, one coming from the number theoretic expression of these numbers and the other coming from their expression as periods on the moduli space of genus 0 curves. By considering relations coming from regularization of non-convergent multizetas, we may define a Hopf algebra of "formal multizetas" whose presentation is given by "extended double shuffle relations". By quotienting this Hopf algebra by products, we obtain a Lie coalgebra, dual to the double shuffle Lie algebra, ds. Although its presentation is understood, the structure of ds remains elusive. . In the recent article, "Derivation and double shuffle relations for multiple zeta values" by K. Ihara, M. Kaneko and Zagier, a structural result on the generators of ds is proved, which relates the depth and weight of multizeta values to associated depth-graded parts of ds. Applied to multizeta values, this result means that whenever the weight and depth of a multizeta value are of opposite parity, then that multizeta value is expressible as a sum, with coefficients in Q, of multizetas of smaller depth and products of multizetas of smaller weight. In my thesis, I prove this result for depths 1 and 2, and the proof gives these coefficients. . In this talk, I will outline the algebraic structure of ds, and discuss the proofs and applications of this paper and this section of my thesis.