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In this paper we investigate the power of migration in online scheduling on multiple parallel machines. The problem is to schedule preemptable jobs with release dates and deadlines on a minimum number of machines. We show that migration, that is, allowing that a preempted job is continued on a different machine, has a huge impact on the performance of a schedule. More precisely, let m be the number of machines required by a migratory solution; then the increase in the number of machines when disallowing migration is unbounded in m. This complements and strongly contrasts previous results on variants of this problem. In both the offline variant and a model allowing extra speed, the power of migration is limited as the increase of number of machines and speed, respectively, can be bounded by a small constant.In this paper, we also derive the first non-trivial bounds on the competitive ratio for non-migratory online scheduling to minimize the number of machines without extra speed. We show that in general no online algorithm can achieve a competitive ratio of f(m), for any function f, and give a lower bound of Omega(log n). For agreeable instances and instances with "loose" jobs, we give O(1)-competitive algorithms and, for laminar instances, we derive an O(log m)-competitive algorithm.