This is a survey of some recent results concerning several classifications of relational structures related to the properties of their self-embedding monoids. For example, if $\preceq^R$ is the right Green's preorder on the monoid $\operatorname{Emb}(\mathbb X)$ of self-embeddings of a structure $\mathbb X$, then the antisymmetric quotient of its inverse is isomorphic to the poset $\mathbb{P(X)}$ of copies of the structure $\mathbb X$ contained in $\mathbb X$ and, defining two structures to be similar if the Boolean completions of the corresponding posets are isomorphic (or, equivalently, if the inverses of the right Green's preorders are forcing-equivalent) we obtain a classification of structures. Some results concerning the posets of copies of specific structures, the interplay between the properties of structures and the properties of their posets of copies, and the corresponding classification of structures and classification of posets representable in this way will be presented.