The convection-driven MHD dynamo in a rotating spherical shell is simulated numerically. Convection cells are regarded as a connecting link between the global and local electromagnetic processes. Local (in many cases, bipolar) magnetic structures are regularly produced by convection cells. Dynamo regimes in “thick” and “thin” shells are discussed. In the first case, the “general” magnetic field maintained by the dynamo has a sign-alternating dipolar component, which varies cyclically, although not periodically. The local structures, as they disintegrate, change into background fields, which drift toward the poles. From time to time, reversals of the magnetic fields in the polar regions occur, as “new” background fields expel the “old” fields. In the second case, the system settles down to a nearly stationary regime without polarity reversals.