Large planar subgraphs in dense graphs

We prove sufficient and essentially necessary conditions in terms of the minimum degree for a graph to contain planar subgraphs with many edges. For example, for all positive γ every sufficiently large graph G with minimum degree at least (2/3+γ)|G| contains a triangulation as a spanning subgraph, whereas this need not be the case when the minimum degree is less than 2|G|/3.