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current sensing is useful in cases where current can be controlled through a few devices as current sources are independent of the resistance through wires. A current source pumping current through 1ohm resistance will be same as 1.1ohm(assuming process variation introduces 0.1ohm of variation). This will not be the case with voltages which will be effected by noise.
current sensing is slow from a speed point of view and requires current sources(which is power hungry). Voltage sensing is faster and has more applications in digital cmos which is more "voltage" based rather than "current".
Current shunts are typically limited to 50 or 75mV to prevent power loss, voltage drop or both and can be integrated into IC current sensors or made discrete with OA differential amps or single ended ground side shunt current sensing.