The study of magnetic anisotropy in metal complexes is at the forefront of current molecular magnetism research because it represents the key property for potential application of molecular materials in spintronics, memory storage, and quantum computing. The anisotropy adds an order of complexity to magnetic properties of complexes, requiring more refined experimental techniques and the use of theoretical tools, especially, of new ab initio approaches for their description. In this review, we discuss the physical reasons for magnetic anisotropy and the mechanisms of its appearance in different metal complexes. Differences in the manifestation of magnetic anisotropy between complexes with weak and strong spin-orbit coupling as well as between single-ion and polynuclear compounds will be emphasized.