I will present properties of polynomials mappings and generalizations. I will first describe all polynomials f and g for wich there is a complex number c such that the orbits c, f(c), F (f(c)), ... and c, g(c), g(g(c)),... have infinite intersection. I will also discuss a common generalization of this result and Mordell’s conjecture (Faltings’ theorem). After this I will move to polynomial mappings over finite fields, with connections to curves having large automorphism groups and instances of a positive characteristic analogue of Riemann’s existence theorem.