This thesis explores various connections between multiple zeta values and modular forms of low level. In the first part, we consider double zeta values of odd weight. We generalize a result of Gangl, Kaneko and Zagier on period polynomial relations among double zeta values of even weights to this setting. This answers a question asked by Zagier. We also prove a conjecture of Zagier on the inverse of a certain matrix in this setting. In the second part, we study multiple zeta values of higher depth. In particular, we give a criterion and a conjectural criterion for "fake" relations in depth 4. In the last part, we consider multiple zeta values of levels 2 and 3. We describe one connection with the Hecke operators T₂ and T₃, and another connection with newforms of level 2 and 3. We also give a conjectural generalization of the Eichler-Shimura-Manin correspondence to the spaces of newforms of levels 2 and 3.