Abstract: In 1994, after extensive numerical calculation, Don Zagier conjectured that $$ \sum_{n_1>n_2>\cdots>n_{2k}>0}\frac{1}{n_1^3n_2n_3^3n_4\cdots n_{2k-1}^3n_{2k}} =\frac{2\pi^{4k}}{(4k+2)!}. $$ This problem remained open until a few years ago when it was settled by the physicist David Broadhurst. At the same time Broadhurst and his co-workers found new conjectures which resisted their methods. In recent joint work with David Bradley, we have settled many of the conjectures of Broadhurst using a certain type of generating function. It turns out that these generating functions are interesting in their own right; for example, these functions are matrix elements in representations of fundamental groups arising in algebraic topology. In a different direction we have obtained partial results towards the so-called ``cyclic insertion conjecture'', a general conjecture about relations between multiple zeta values. This talk will survey some of the above topics.