Conway's field No of surreal numbers comes both with a natural total order and an additional "simplicity relation" which is also a partial order. Considering No as a doubly ordered structure for these two orderings, an isomorphic copy of No into itself is called a surreal substructure. It turns out that many natural subclasses of No are actually of this type. In this paper, we study various constructions that give rise to surreal substructures and analyze important examples in greater detail.