Given the extent and importance of financial interconnectedness in recent years that were particularly underlined by the 2007-2009 financial crisis, the adoption of the network paradigm to analyze and improve robustness of a financial system appears to be fully relevant. Financial institutions are viewed as nodes of a network and their short- or long-term loans extended to each other as links or exposures through which a shock may propagate. Moreover, the same crisis accentuated the role of funding shortage as a channel of shock transmission. This dissertation focuses on the interplay of liquidity stress, interbank contagion and a network structure with application to the European interbank market and payment system. The contribution of this research to the literature on financial networks is threefold. The first develops a model that allows analyzing three contagion channels that happened to be at play during the financial crisis: exposures to a common risk factor; exposures to credit and counterparty risk in the interbank market; exposures to short-term liquidity risk. The second contribution is the unique analysis of cross-border contagion in the European banking system from 2008 to 2012 at the bank level using the developed model. Overall, the study finds the importance of the network structure for the extent of contagion propagation and captures the fragmentation of the market observed in 2011-2012. The third contribution consists of analysis of payment delays in the European payment system TARGET2. More specifically, this chapter provides evidence that banks differ in the way they manage their daily liquidity and can be split into two groups in this regard: those which put enough initial liquidity into the system, and those which economize on liquidity and rely on incoming payments to make outgoing transactions. The second group is responsible for the majority of the delayed payments, particularly during the period of low liquidity in the market, which constitutes an early warning indicator of stress.