The existence of a dominant massive planet, Jupiter, in our solar system, although perhaps essential for long-term dynamical stability and the development of life, may not be typical of planetary systems that form around other stars. In a system containing two Jupiter-like planets, the possibility exists that a dynamical instability will develop. Computer simulations suggest that in many cases this instability leads to the ejection of one planet while the other is left in a smaller, eccentric orbit. In extreme cases, the eccentric orbit has a small enough periastron distance that it may circularize at an orbital period as short as a few days through tidal dissipation. This may explain the recently detected Jupiter-mass planets in very tight circular orbits and wider eccentric orbits around nearby stars.