How many satellites have been discovered in the Solar System after Galileo?

By the beginning of 2010 the total number of natural satellites and multiple systems in the Solar System was equal to 350, including: 168 satellites of large planets, 119 multiple asteroids (including main-belt and near-Earth asteroids, Mars-crossers and Jupiter Trojan asteroids) and 63 multiple transneptunian and Kuiper-belt objects. Meanwhile, we cannot count precisely how many moons in total have been discovered to date due to the deficiency of accepted definitions.