We discuss the dynamics of exponential maps $z\mapsto \lambda e^z$ from the point of view of dynamic rays, which have been an important tool for the study of polynomial maps. We prove existence of dynamic rays with bounded combinatorics and show that they contain all points which ``escape to infinity'' in a certain way. We then discuss landing properties of dynamic rays and show that in many important cases, repelling and parabolic periodic points are landing points of periodic dynamic rays. For the case of postsingularly finite exponential maps, this needs the use of spider theory.