Where do comets come from and where do they go?
Comets are believed to have two sources. Long-period comets (those which take more than 200 years to complete an orbit around the Sun) originate from the Oort Cloud. Short-period comets (those which take less than 200 years to complete an orbit around the Sun) originate from the Kuiper Belt.
What eventually happens to comets?
COMETS vaporize when their orbits take them close to the Sun. ... After many orbits near the Sun, a comet does eventually "expire." In some cases, all the volatile ices boil away, leaving a remnant of rock and dust. Sometime the comet completely disintegrates.