  Rapid developments in genetics and biology have led to phylogenetic methods
becoming an important direction in the study of cancer and viral evolution.
Although our understanding of gene biology and biochemistry has increased and
is increasing at a remarkable rate, the theoretical models of genetic evolution
still use the phylogenetic tree model that was introduced by Darwin in 1859 and
the generalization to phylogenetic networks introduced by Grant in 1971.
Darwin's model uses phylogenetic trees to capture the evolutionary
relationships of reproducing individuals [6]; Grant's generalization to
phylogenetic networks is meant to account for the phenomena of horizontal gene
transfer [14]. Therefore, it is important to provide an accurate mathematical
description of these models and to understand their connection with other
fields of mathematics. In this article, we focus on the graph theoretical
aspects of phylogenetic trees and networks and their connection to stable
curves. We introduce the building blocks of evolutionary moduli spaces, the
dual intersection complex of the moduli spaces of stable curves, and the
categorical relationship between the phylogenetic spaces and stable curves in
$\overline{\mathfrak{M}}_{0,n}(\mathbb{C})$ and
$\overline{\mathfrak{M}}_{0,n}(\mathbb{R})$. We also show that the space of
network topologies maps injectively into the boundary of
$\overline{\mathfrak{M}}_{g,n}(\mathbb{C})$.
