I consider the following question: what aspects of the geometry of a variety prohibit the existence of an "interesting" dynamical system on that variety?  There are various geometric conditions on a variety which forbid the existence of algebraically or arithmetically interesting dynamical systems.  I will discuss various conditions of this kind, and the consequences for conjectures regarding unlikely intersections in algebraic dynamics. Time permitting, I will discuss recent work with Paul Reschke on the non-existence of polarized endomorphisms for varieties with non-surjective Albanese map, and consider what kind of endomorphisms these varieties can carry.