Despite the chaotic nature of the three-body problem, exact periodic solutions exist — orbits that repeat their trajectory after a fixed time interval. The main known families are listed below.

A family discovered in the late 20th century. The system does not rotate as a whole — total angular momentum is zero. The most famous representative is the figure-eight orbit.

The most famous periodic solution, discovered by Moore (1993) and popularized by Chenciner and Montgomery (2000). Three equal-mass bodies trace a figure-eight trajectory, alternately catching up to each other. Period ~6.4 time units, stable to small perturbations.

In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović systematically explored the initial condition space and found new classes of periodic orbits with equal masses. The names describe the visual shape of the trajectory:

• Butterfly I — the trajectory resembles butterfly wings. One of the most aesthetic solutions.
• Dragonfly — an elongated orbit with characteristic lateral wings.
• Moth I—III — three varieties differing in the number of trajectory lobes.
• Goggles — two symmetric loops connected by a bridge.
• Yin-Yang — an orbit with inner and outer parts, resembling the yin-yang symbol.
• Bumblebee — a chaotic-looking yet strictly periodic orbit with buzzing motion.
• Yarn — a complex intertwined trajectory resembling a ball of yarn.

Sheen orbit families (2016): include ovals, ring shapes, and new topologies, including Yarn-ovals. Shuttle-type orbits: one body oscillates back and forth between the other two, resembling a shuttle in a loom. Animated examples of such complex systems can be found in scientific discussions (Reddit, arXiv).

When one body has negligible mass (satellite, asteroid), the problem reduces to the motion of a test particle in the field of two massive bodies.

Planar periodic orbits around the L₁ and L₂ libration points. Used for spacecraft maneuvers.

Three-dimensional periodic orbits around L₁, L₂. The James Webb telescope is on a halo orbit around the Sun—Earth L₂ point.

Quasi-periodic orbits describing complex shapes resembling Lissajous curves — pendulum oscillations in different planes.