Optical modulators have been and will continue to be essential devices for energy- and cost-efficient optical communication networks. Heterogeneous silicon and lithium niobate modulators have demonstrated promising performances of low optical loss, low drive voltage, and large modulation bandwidth. However, DC bias drift is a major drawback of optical modulators using lithium niobate as the active electro-optic material. Here, we demonstrate high-speed and bias-drift-free Mach–Zehnder modulators based on the heterogeneous silicon and lithium niobate platform. The devices combine stable thermo-optic DC biases in silicon and ultra-fast electro-optic modulation in lithium niobate, and exhibit a low insertion loss of 1.8 dB, a low half-wave voltage of 3 V, an electro-optic modulation bandwidth of at least 70 GHz, and modulation data rates up to 128 Gb/s.