The rich and varied ways that genetic material can be passed between species has motivated extensive research into the theory of phylogenetic networks. Features that align with biological processes, or with desirable mathematical properties, have been used to define classes and prove results, with the goal of developing the theoretical foundations for network reconstruction methods. We may have now reached the point where a collection of recent results can be drawn together to make one class of network, the \emph{normal} networks, a leading contender, sitting in the sweet spot between biological relevance and mathematical tractability.