The 2007-2008 global financial crisis has been associated with a high level of connectivity in the global financial system. The crisis, and the following events of the past decade, have highlighted the relevance of the concept of interconnectedness to understanding systemic risk, transmission of financial contagion and ultimately on the subject of financial stability. Nevertheless, the more general relationship, across its full spectrum, between interconnectedness and financial stability, is still not fully studied and understood. This paper reviews the positive aspects as well as the negative aspects of interconnectedness. It also discusses briefly the important question of the optimal level of connectivity in a financial system. Finally, the paper proposes the use of novel statistical inferential methods for complex networks to address comprehensively the study of interconnectedness in financial systems.