The Strominger-Yau-Zaslow (SYZ) proposal, although not stated very precisely, is now understood to be overly optimistic. One must allow for various degenerations and singularities; even so, there is still no single precise form of the SYZ conjecture. Nonetheless, its conceptual picture has been enormously influential in the study of mirror symmetry, and research on its various facets is currently an active field. It can be contrasted with an alternative (and equally influential) proposal by Maxim Kontsevich known as homological mirror symmetry, which deals with purely algebraic structures.