<p>Understanding the patterns and processes of cophylogeny and coevolution has long been of high importance in evolutionary ecology despite theoretical and methodological challenges. As widespread hybridization and histories of introgression become increasingly clear across the tree of life, biologists must contend with the methodological and theoretical implications. Nowhere is the challenge clearer than in the field of cophylogeny, where current approaches are fundamentally dependent on bifurcating trees, and evolutionary thinking often has neglected network perspectives. From plant-pollinator mutualisms to host-parasite antagonisms, contemporary hybridization, introgression, and reticulation (“network evolution”) are present in cophylogenetic and coevolutionary systems, yet much research assumes these processes are absent or negligible. We argue here that network evolutionary processes in cophylogenetic and coevolutionary systems are not just nuisances, but may be regulators of relationships, causing acceleration, disruption, or prevention of potential coevolutionary dynamics. In the face of continued recognition that network evolution is a significant force across the tree of life, specific theories, methodologies, and hypotheses on the result of interactions between network evolution and cophylogeny or coevolution must be developed, three of which are advanced here. First, we hypothesize that hybrid host individuals may act as phylogenetic “bridges,” facilitating symbiont transfer between otherwise reproductively isolated lineages. Second, we hypothesize that coevolutionary dynamics and network evolution may form feedback loops, wherein coevolutionary trajectories elevate introgression rates, and introgression in turn alters the coevolutionary landscape. Third, we hypothesize that introgression may act as an accelerator of coevolution, analogous to known effects of horizontal gene transfer and population mixing. We explore these questions in two relatively well-studied coevolutionary systems that involve extensive histories of introgression and contemporary hybrid lineages. We also suggest additional hypotheses in systems where hybridization in symbiotic lineages and cophylogenetic or coevolutionary dynamics may influence one another. Finally, we discuss the methodological and theoretical challenges of incorporating network evolutionary thinking into the fields of cophylogenetics and coevolution.</p>