Mostly comets follow elliptical orbits just like the planets, except more elongated (eccentricity higher). A few comets that come from deep space are on hyperbolic orbits and are very fast-moving, they visit once and never come back because they have enough kinetic energy to escape again.
Comets travel in highly elliptical orbits. See link for more.
Comets
Comets. Periodic comets have highly eccentric elliptical orbits, with eccentricities.
Comets are balls of ice and dust in orbit around the Sun. The orbits of comets are different from those of planets - they are elliptical. A comet's orbit takes it very close to the Sun and then far away again.
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The Oort Cloud
All orbits are geodesic curves. Comets tend to have elliptical orbits ... as do planets, really; the degree of eccentricity (this is a measure of how "stretched" the ellipse is) just tends to be higher for comets.
We generally believe that comets with orbits of 200 years or less, traveling in the ecliptic plane, come from the Kuiper Belt. Those comets with longer orbits, or that travel at angles to the ecliptic plane, come from the Oort Cloud.
elongated elliptical orbits around the sun
Oort cloud comets
Comets do indeed have orbits. Sometimes the orbits take thousands of years. For the most part, they stay in these orbits unless affected by the gravity of another body.
Comets go around the Sun in a highly elliptical orbit.