This talk will be devoted to studying what can be said about the multizeta values in the Lie algebra situation, i.e. instead of considering the Q-algebra of multizeta values, we will consider the vector space obtained by quotienting it by "old MZV"'s, i.e. products of MZV's, keeping only the "new" ones, which are multiplicative generators of the algebra. The goal is to determine all algebraic relations between MZV's. In order to avoid basic transcendence problems, we will formalize the MZV's, i.e. make the assumption that they are all transcendent. Under this assumption, there are at last four different conjectural combinatorial descriptions of the vector space of new MZV's, all of which are not merely vector spaces, but Lie coalgebras. The talk will be devoted to describing and comparing the four conjectured Lie coalgebras (one is free, another is the Lie algebra of GT...) and showing the surprising number theoretic elements (modular forms, Bernoulli numbers) that arise when studying their structure.