In this review we focus on the fundamental theory of magnetohydrodynamic reconnection, together with applications to understanding a wide range of dynamic processes in the solar corona, such as flares, jets, coronal mass ejections, the solar wind and coronal heating. We summarise only briefly the related topics of collisionless reconnection, non-thermal particle acceleration, and reconnection in systems other than the corona. We introduce several preliminary topics that are necessary before the subtleties of reconnection can be fully described: these include null points (Sects. 2.1–2.2), other topological and geometrical features such as separatrices, separators and quasi-separatrix layers (Sects. 2.3, 2.6), the conservation of magnetic flux and field lines (Sect. 3), and magnetic helicity (Sect. 4.6). Formation of current sheets in two- and three-dimensional fields is reviewed in Sect. 5. These set the scene for a discussion of the definition and properties of reconnection in three dimensions that covers the conditions for reconnection, the failure of the concept of a flux velocity, the nature of diffusion, and the differences between two-dimensional and three-dimensional reconnection (Sect. 4). Classical 2D models are briefly presented, including magnetic annihilation (Sect. 6), slow and fast regimes of steady reconnection (Sect. 7), and non-steady reconnection such as the tearing mode (Sect. 8). Then three routes to fast reconnection in a collisional or collisionless medium are described (Sect. 9). The remainder of the review is dedicated to our current understanding of how magnetic reconnection operates in three dimensions and in complex magnetic fields such as that of the Sun’s corona. In Sects. 10–12, 14.1 the different regimes of reconnection that are possible in three dimensions are summarised, including at a null point, separator, quasi-separator or a braid. The role of 3D reconnection in solar flares (Sect. 13) is reviewed, as well as in coronal heating (Sect. 14), and the release of the solar wind (Sect. 15.2). Extensions including the role of reconnection in the magnetosphere (Sect. 15.3), the link between reconnection and turbulence (Sect. 16), and the role of reconnection in particle acceleration (Sect. 17) are briefly mentioned.