Abstract: I study 6 monoids which may be naturally associated with any first order structure (and may be with other kinds of structures such as topological spaces), comprising all endomorphisms, epimorphisms, monomorphisms, embeddings, bimorphisms and automorphisms respectively. For the random graph for instance, these monoids are all distinct, though for other structures some of them may coincide. The main themes are: corresponding versions of "homomorphism-homogeneity", generalizing the classical notion of homogeneity, for instance, by requiring that all homomorphisms from a finite substructure to the structure extend to epimorphisms (which follows up an idea of Cameron and Nesetril), the notion of a generic member of one of the monoids, more details about a particular case, the rationals as an ordered set, and concluding with some remarks on Sierpinski type theorems.