An important invariant of an automorphism f of an algebraic variety X is its growth\, i.e.\, how does the norm of the inverse image operator on the cohomology group H2(X\,C) of the n'th power fn depend on n. The growth can be exponential\, polynomial\, or bounded. There are many beautiful results describing constraints on the geometry of X admitting an automorphism with exponential or polynomial growth. We are going to discuss the geometric properties of a variety X which admits an infinite order automorphism with bounded growth. It appears there are only two sources of these automorphisms; namely\, automorphisms of projective spaces and translations on abelian varieties. In particular\, a rationally connected threefold whose automorphism group contains an element f of infinite order with bounded growth is necessarily a rational variety\, and an iterate of the automorphism f is birationally conjugate to a regular automorphism of the projective space. I am going to explain how to work with these automorphisms in any dimension and to prove this result.