In a seminal 1972 paper, Robert M. May asked: “Will a Large Complex System Be Stable?” and argued that stability (of a broad class of random linear systems) decreases with increasing complexity, sparking a revolution in our understanding of ecosystem dynamics. Twenty-five years later, May, Levin, and Sugihara translated our understanding of the dynamics of ecological networks to the financial world in a second seminal paper, “Complex Systems: Ecology for Bankers.” Just a year later, the US subprime crisis led to a near worldwide “great recession,” spread by the world financial network. In the present paper we describe highlights in the development of our present understanding of stability and complexity in network systems, in order to better understand the role of networks in both stabilizing and destabilizing economic systems. A brief version of this working paper, focused on the underlying theory, appeared as an invited feature article in the February 2020 Society for Chaos Theory in Psychology and the Life Sciences newsletter (Hastings et al. 2020).