Magnetic shielding and magnetic cores are two sides of the same coin.
Permalloy is noted for its very high magnetic permeability. When this property is used to remove magnetism from those sections which are being affected adversely by magnetism and guide it to unaffected areas, it is known as magnetic shielding. Meanwhile, when it is used to capture weak magnetic signals, it becomes a magnetic core.
Any textbook on electromagnetism will tell you that electrical current and magnetism are closely related, in that fluctuations in the magnetic field generate electrical current (Lenz's law), and that electrical current in turn generates a magnetic field (Ampère's circuital law). There are technologies for measuring electric current which utilize this relationship well.
Clamp meters for electric current are a typical example of these. Among clamp meters requiring a high degree of sensitivity, there are some which use permalloy cores.
Current transformers are devices used to increase and decrease the current value, and are effective for such things as lowering excessive electric current on the primary side to a smaller, more manageable level on the secondary side. They are similar to transformers used to increase and decrease voltage.
Some high sensitivity CT employ permalloy cores. .