A system may include one or more finger-mounted devices such as finger devices with U-shaped housings configured to be mounted on a user's fingers while gathering sensor input and supplying haptic output. The finger devices may have power receiving circuitry configured to receive power from a power source. The power source may be incorporated into an electronic device such as a battery case, a head-mounted display, or a wireless charging mat or stand. The power source may supply power through terminals that form ohmic contacts with mating terminals in the finger device or may transmit power wirelessly using capacitive coupling or inductive charging arrangements. A finger device may have hinge structures that allow portions of the device to rotate relative to each other.