The burning number of a graph was recently introduced by Bonato et al. Although they mention that the burning number generalises naturally to directed graphs, no further research on this has been done. Here, we introduce graph burning for directed graphs, and we study bounds for the corresponding burning number and the hardness of finding this number. We derive sharp bounds from simple algorithms and examples. The hardness question yields more surprising results: finding the burning number of a directed tree is NP-hard, but FPT; however, it is W[2]-complete for DAGs. Finally, we give a fixed-parameter algorithm to find the burning number of a digraph, with a parameter inspired by research in phylogenetic networks.