Sum Theorems for Multiple Zeta Values, Old and New

One of the first interesting results about multiple zeta values to
be noticed was that the sum of all convergent multiple zeta values
of fixed weight and depth is simply zeta of the weight.  This result
has since been generalized in many ways.  We will talk about
several sum theorems of recent interest (including one that is
still conjectural) and an appropriate algebraic framework for
them.