There are millions of asteroids within the solar system. As of May 2022 there are 614,690 numbered minor planets, and >500,000 un-numbered. Most of the un-numbered asteroids are known to exist but require further observation before they receive official numbered status. The vast majority of the numbered and un-numbered asteroids reside within the main belt of asteroids, between the orbits of Mars and Jupiter; far fewer of the these asteroids are Near-Earth asteroids, some of which have orbits that intersect Earth's orbit and also dip into the main belt.
It is estimated that there are between 700,000 and 1,700,000 asteroids equal or larger than 1 kilometer across. If we include all possible asteroids down to only a meter across, there are likely 100s of millions out there! The smaller the asteroid, the larger the population.
A subset of all asteroids are the Near-Earth asteroids (NEAs). As of May 2022 there are > 29,000 known NEAs, and asteroid surveys are adding >2500 new NEAs each year.