The most traditional configuration space C(X,n) of a complex space X consists of all n point subsets ("configurations") Q in X.

If X carries an additional geometric structure, it may be taken into account; say if X=CP^m or C^m, the space C(X,n;gp) of geometrically generic configurations consists of all n point configurations Q such that no hyperplane in X contains more than m points of Q. An automorphism T of X (preserving an additional geometric structure, whenever it is relevant) produces a holomorphic endomorphism f of the configuration space via f(Q)=TQ. If the automorphism group Aut X is a complex Lie group, one may take T=T(Q) depending analytically on a configuration Q and define the corresponding holomorphic endomorphism f by f(Q)=T(Q)Q. Such a map f is called tame. In the talk, we shall see that for every non-hyperbolic Riemann surface X all "non-degenerate" holomorphic endomorphisms of configuration spaces C(X,n) are tame. To some extent, this is true also for spaces of geometrically generic configurations.