Toroids:

Inductor coils do not have to be wound on cylindrical forms, or on cylindrical ferromagnetic cores. In recent years, a new form of coil has become increasingly common. This is the toroid. It was named from the donut shape of ferromagnetic core. The coil is wound over the core having this shape.

Figure--   Permeability tuning of a solenoidal coil.

Figure--   A toroidal coil winding.

There are several advantages to toroidal coils over solenoidal, or cylindrical, ones. 1) Fewer turns of wire are needed to get a certain inductance with the toroid, as compared with the solenoid.
2) A toroid can be physically smaller for the given inductance and current carrying capacity.
3) The most important, essentially all of the flux in a toroidal inductor is contained within the core material. This reduces the unwanted mutual inductances with components near toroid.

There are some disadvantages in the flexibifity, of toroidal coils. It is difficult to permeability tune a toroidal coil than it is to tune a solenoidal one. It is been done, but the hardware is cumbersome. Toroidal coils are difficult to wind than so lenoidal ones.

At times, mutual inductance between or among physically separate coils is wanted; with a toroid, the coils have to be wound on the same form for this to be possible.