Four hundred years ago Galileo turned his telescope to the heavens and changed the way we view the cosmos forever. Among his discoveries in January of 1610 were four new ‘stars’, following Jupiter in the sky but changing their positions with respect to the giant planet every night. Galileo showed that these ‘Medicean stars’, as he named them, were moons orbiting Jupiter in the same manner that the Earth and planets revolve about the Sun in the Copernican theory of the solar system. Over the next three centuries these moons, now collectively named the Galilean satellites after their discoverer, remained tiny dots of light in astronomers' telescopes. In the latter portion of the twentieth century Galileo's new worlds became important targets of exploration by robotic spacecraft. This paper reviews the history of this exploration through the discoveries made by the Galileo mission from 1995 to 2003, setting the stage for on-going exploration in the new century.