The Orb

a.k.a. Orb-It, l'ORBS
Patented by C. Wiggs and C. Taylor 1985, made by Parker Brothers.
(plastic, 3.5 inches)

When solved, the north pole ring has 8 yellow, the upper equator 20 green, the lower equator 20 red, and the south pole 8 blue beads. The puzzle can twist in 45 degree increments. The 45 (also -45, 135, and -135) degree rotations give one continuous loop along which all beads can rotate; fun, but the beads are hard to move this way, and it is not needed to solve the puzzle. A 90 (and -90) degree rotation creates two cycles of 28 beads that connect halves of the pole and equator rings. The 180 degree rotation joins opposite halves of the pole and equator rings.
Based on the observation that incorrect beads always occur in multiples of 2, a simple solution, which is presented on Jaap's Page, works in three steps:
1. Use 90 deg. twists to get yellow/blue to poles, green/red to equator.
2. Use 180 deg. twists to get north yellow, south blue.
3. Use 180 deg. twists to get upper equator green, lower equator red.