The detection of extrasolar planets is one of the great scientific discoveries of the past decade. Most of these planets planets move on orbits with substantial eccentricities. The origin of these large eccentricities is an unsolved puzzle. We propose that they result from the exchange of angular momentum and energy between the planets and the disks from which they form. These interactions are concentrated at discrete Lindblad and corotation resonances. We describe the physics of these resonances and their effects on the planets migration and eccentricity evolution. If both resonances are fully active, the rate of eccentricity damping by corotation resonances is slightly larger than its excitation rate by Lindblad resonances and the eccentricity decays. However, corotation resonances tend to change the properties of the disk, and therefore partially saturate. This tips the balance in favor of eccentricity excitation by Lindblad resonances. Sufficient saturation is possible only if the initial eccentricity is higher than a few percent. This minimal eccentricity is required to overcome competition with viscosity which acts to desaturate the resonances. Thus eccentricity growth is a finite amplitude instability.