Double zeta values and modular forms.

We give new relations among double zeta values and show that the structure of the Q-vector space of all (known) relations among double zeta values of weight k is connected in many different ways with the structure of the space of modular forms of weight k on the full modular group. Furthermore, we introduce and study both transcendental and combinatorial ``double Eisenstein series'' which explain the relation between double zeta values and modular forms and provide new realizations of the space of double zeta relations