Introgression is an evolutionary process in which genes or other types of genetic material are introduced into a genome. It is an important evolutionary process that can, for example, play a fundamental role in speciation. Recently the concept of an overlaid species forest was introduced to represent introgression histories. Basically this approach takes a putative gene history in the form of a phylogenetic gene tree and tries to overlay this onto a forest which usually consists of a collection of lineage trees for the species of interest. The result is a network called an overlaid species forest in which genes jump or introgress between lineages. In this paper we study properties of overlaid species forests, showing that they have various connections with models for lateral gene transfer, maximum parsimony, and unfolding of phylogenetic networks. In particular, we show that a certain algorithm called OSF-B UILDER for constructing overlaid species forests is guaranteed to a produce a special type of overlaid species forest with a minimum number introgressions, as well as providing some characterizations for networks that can arise from overlaid species forests. We expect that these results will be useful in developing new methods for representing introgression histories, a growing area of interest in phylogenetics.