The number of moons is constantly changing for some planets as they are discovered, but at the last official count there are 166 moons, with more than 20 waiting to be recognized as moons. This count is only of the moons that orbit recognized planets in the solar system. There are numerous other satellites, like asteroids, that have what would be considered moons circling them as well.
Jupiter and Saturn have the highest number of moons out of the recognized planets. Jupiter has 63, with more waiting to be recognized as moons. Saturn also has around 60, with more waiting to be added to the list. The planets with the least number of moons are Mercury and Venus, with no moons at all.