Analyzing the qualitative three-dimensional magnetic structure of a plasmoid, we were led to reconsider the concept of magnetic reconnection from a general point of view. The properties of relatively simple magnetic field models provide a strong preference for one of two definitions of magnetic reconnection that exist in the literature. Any concept of magnetic reconnection defined in terms of magnetic topology seems naturally restricted to cases where the magnetic field vanishes somewhere in the nonideal (diffusion) region. The main part of this paper is concerned with magnetic reconnection in nonvanishing magnetic fields (finite-B reconnection), which has attracted less attention in the past. We show that E∥ (the electric field component parallel to the magnetic field) plays a crucial physical role in finite-B reconnection, and we present two theorems involving E∥. The first states a necessary and sufficient condition on E∥ for global reconnection to occur. Here the term ''global'' means the generic case where the breakdown of magnetic connection occurs for plasma elements that stay outside the nonideal region. The second theorem relates the change of magnetic helicity to E∥ for cases where the electric field vanishes at large distances. That these results provide new insight into three-dimensional reconnection processes is illustrated in terms of the plasmoid configuration, which was our starting point. ¿ American Geophysical Union 1988