We introduce the notion of stationary Apollonian packings in the d-dimensional Euclidean space  as a mathematical formalization of random Apollonian packings and rotational random Apollonian packings, which constitute popular grain packing models in physics. Apart from dealing with issues of existence and uniqueness in the entire Euclidean space,  we provide asymptotic results for the growth durations and show that the packing is almost surely space-filling. Finally, we observe that grains arrange in clusters and investigate properties related to percolation. This talk is based on joint work with Gary Delaney and Volker Schmidt.