  This thesis is a study of algebraic and geometric relations between multizeta
values. In chapter 2, we prove a result which gives the dimension of the
associated depth-graded pieces of the double shuffle Lie algebra in depths 1
and 2. In chapters 3 and 4, we study geometric relations between multizeta
values coming from their expression as periods on $\mathfrak{M}_{0,n}$. The key
ingredient in this study is the top dimensional de Rham cohomology of special
partially compactified moduli spaces associated to multizeta values. In chapter
3, we give an explicit expression for a basis, represented by polygons, of this
cohomology. In chapter 4, we generalize this method to explicitly describe the
bases of the cohomology of other partially compactified moduli spaces. This
thesis concludes with a result which gives a new presentation of
$Pic(\overline{\mathfrak{M}}_{0,n})$.
