The Fulton-MacPherson conguration space is a natural compactication of the conguration space of n ordered points on a smooth projective variety . The Kontsevich moduli space parametrizing stable maps from n-pointed rational curves to a projective space is another widely studied algebraic variety, and plays a central role in algebraic geometry, string theory and Gromov-Witten theory. These two spaces are indeed closely related. In this seminar we will discuss properties of fibrations of these spaces, and compute their automorphism groups.