We study the internal structure and the convective motions of giant extrasolar planets in order to calculate their dynamo-generated surface magnetic field and dipolar magnetic moment. Using evolutionary models, we investigate the existence of magnetism in planets with masses ranging from 0.3MJ to 10MJ and with rotation periods ranging from synchronism in "hot Jupiters" (with periods of 3-4 days) to the breakdown centrifugal limit (with periods of 2-5 hr). We find that the high Rayleigh and low Ekman numbers in the interior metallic-hydrogen region guarantee convective motions even for low-mass and evolved (aged) planets. The convective velocities estimated from mixing-length theory and from "magnetostrophic" balances yield high Reynolds and magnetic Reynolds numbers (~103-106), permitting the self-excited dynamo mechanism. Strong magnetism occurs in young massive and rapidly rotating extrasolar planets (with surface magnetic fields of ~30-60 G), but older or orbitally synchronized planets should have fields of ~1 G. We discuss the implications of these results for the detection of magnetic fields in extrasolar planets.