Since the emergence of optical fiber communications, lithium niobate (LN) has been the material of choice for electro-optic modulators, featuring high data bandwidth and excellent signal fidelity. Conventional LN modulators however are bulky, expensive and power hungry, and cannot meet the growing demand in modern optical data links. Chip-scale, highly integrated, LN modulators could offer solutions to this problem, yet the fabrication of low-loss devices in LN thin films has been challenging. Here we overcome this hurdle and demonstrate monolithically integrated LN electro-optic modulators that are significantly smaller and more efficient than traditional bulk LN devices, while preserving LN's excellent material properties. Our compact LN electro-optic platform consists of low-loss nanoscale LN waveguides, micro-ring resonators and miniaturized Mach-Zehnder interferometers, fabricated by directly shaping LN thin films into sub-wavelength structures. The efficient confinement of both optical and microwave fields at the nanoscale dramatically improves the device performances featuring a half-wave electro-optic modulation efficiency of 1.8 V∙cm while operating at data rates up to 40 Gbps. Our monolithic LN nanophotonic platform enables dense integration of high-performance active components, opening new avenues for future high-speed, low power and cost-effective communication networks.