Suction cups are the light bulbs of robotics and automation. They are simple, reliable, yet energy-hungry, and require a bulky and noisy vacuum infrastructure. This work reports Electroadhesion (EA) Suction Cups: soft, silent, monolithic, electrically-driven grippers, with a power consumption of only 1.5 W, that can grasp flat and curved objects, with smooth or rough surfaces, holding payloads up to 1.5 kg. This performance is enabled by a deeper understanding of the contact mechanics of electroadhesion systems. A thin and soft membrane containing interdigitated electrodes zips onto the object driven by electrostatic forces, conforming to the object's shape and thus establishing large-area contact. The lifting force is transmitted to a robot arm through a small pillar connected at the center of the membrane. This design maximizes the peeling force and enables the formation of passive vacuum inside the conical chamber formed when the membrane stretches during lifting. Object release is obtained by turning off the voltage and optionally by opening a valve to quickly break the vacuum. EA suction cups address many shortcomings of widely used vacuum-driven grippers, offering a compact, fully electric, and energy-efficient solution that meets the needs for efficiency and portability in both industrial and service robotics.