  Phylogenetic networks extend phylogenetic trees to allow for modeling
reticulate evolutionary processes such as hybridization. They take the shape of
a rooted, directed, acyclic graph, and when parameterized with evolutionary
parameters, such as divergence times and population sizes, they form a
generative process of molecular sequence evolution. Early work on computational
methods for phylogenetic network inference focused exclusively on reticulations
and sought networks with the fewest number of reticulations to fit the data. As
processes such as incomplete lineage sorting (ILS) could be at play
concurrently with hybridization, work in the last decade has shifted to
computational approaches for phylogenetic network inference in the presence of
ILS. In such a short period, significant advances have been made on developing
and implementing such computational approaches. In particular, parsimony,
likelihood, and Bayesian methods have been devised for estimating phylogenetic
networks and associated parameters using estimated gene trees as data. Use of
those inference methods has been augmented with statistical tests for specific
hypotheses of hybridization, like the D-statistic. Most recently, Bayesian
approaches for inferring phylogenetic networks directly from sequence data were
developed and implemented. In this chapter, we survey such advances and discuss
model assumptions as well as methods' strengths and limitations. We also
discuss parallel efforts in the population genetics community aimed at
inferring similar structures. Finally, we highlight major directions for future
research in this area.
