Highly-integrated galvanically isolated current sensing solutions are required to satisfy the demands arising from recent developments in areas such as electric vehicles, smart homes and industrial electronics. These sensors must also be able to provide measurements for wide bandwidths and with high sensitivity levels according to their application. In this thesis, a wide-bandwidth and highly sensitive fully integrated shunt-resistor current sensor for off-chip current sensing is presented. The architecture of the proposed sensor includes a voltage-controlled oscillator-based frequency modulator that offers inductive galvanic isolation and a phase-locked loop-based frequency demodulator. This current sensor solution is implemented in a 65nm CMOS technology, occupies an area of 1.88 mm^2, and achieves a bandwidth of 12.5 MHz. Realizing a sensitivity of 19 mV/A, this sensor is one of the most sensitive solutions reported in the state of the art.