In this second part of the book we develop the theoretical foundations for phylogenetic networks. First we study splits and unrooted phylogenetic networks (Chapter 5). The mathematical and computational aspects of such networks have been worked out in detail over the last 20 years. The existing algorithms are routinely used in numerous publications in systematic biology. Then we turn to clusters and rooted phylogenetic networks (Chapter 6). Their theory is still under development and widely used methods for the computation of rooted phylogenetic networks have yet to emerge.