Strong tidal interaction with the central star can circularize the orbits of close-in planets. With the standard tidal quality factor Q of our solar system, estimated circularization times for close-in extrasolar planets are typically shorter than the ages of the host stars. While most extrasolar planets with orbital radii a ?= 0.1 AU indeed have circular orbits, some close-in planets with substantial orbital eccentricities have recently been discovered. This new class of eccentric close-in planets implies that either their tidal Q factor is considerably higher, or circularization is prevented by an external perturbation. Here we constrain the tidal Q factor for transiting extrasolar planets by comparing their circularization times with accurately determined stellar ages. Using estimated secular perturbation timescales, we also provide constraints on the properties of hypothetical second planets exterior to the known ones.