Reconstructing the avian tree of life has become one of the major goals in ornithology. The use of genomic tools seemed a promising approach to reach this goal, but, instead, phylogenetic analyses of large numbers of genes uncovered high levels of incongruence between the resulting gene trees. This incongruence can be caused by several biological processes, such as recombination, hybridization, and rapid speciation (which can lead to incomplete lineage sorting). These processes directly or indirectly amount to deviations from tree-like patterns, thereby thwarting the use of phylogenetic trees. Phylogenetic networks provide an ideal tool to deal with these difficulties. We illustrate the usefulness of phylogenetic networks to capture the complexity and subtleties of diversification processes by discussing several recent genomic analyses of birds in general and the well-known radiation of Darwin's finches. With the increasing amount of genomic data in avian phylogenetic studies, capturing the evolutionary history of a set of taxa in a phylogenetic tree will become increasingly difficult. Moreover, given the widespread occurrence of hybridization and the numerous adaptive radiations in birds, phylogenetic networks provide a powerful tool to display and analyse the evolutionary history of many bird groups. The genomic era might thus result in a paradigm shift in avian phylogenetics from trees to bushes.