The measurement of the electric current can be classified depending upon the underlying fundamental physical principles such as,
Faraday's Law of Induction
Magnetic field sensors
Faraday effect
Transformer or current clamp meter, (suitable for AC current only).
Fluxgate sensor, (suitable for AC or DC current).
Hall effect sensor(suitable for AC, DC, or pulsating current), a type of current sensor which is based on the Hall Effect phenomenon discovered by Edwin Hall in 1879.
Shunt resistor, whose voltage is directly proportional to the current through it.
Fiber optic current sensor, using an interferometer to measure the phase change in the light produced by a magnetic field.
Rogowski coil, electrical device for measuring alternating current (AC) or high speed current pulses.
Giant Magnetoresistance(GMR): Magnetic field sensor suitable for AC & DC Current with higher accuracy than Hall Effect. Placed parallel to the magnetic field.

Imagine if a BIG LM317 in current mode, got rid of the resistor, and used a magic box across the bus bar connected to the 'sense' pin.