Payments and financial market infrastructures are vital components of modern economies, yet they have often been overlooked by economists. However, any economic exchange, including financial market transactions, is matched by a payment leg, and the safety, efficiency, and immediacy of the payment, and the problem of linking the two legs of the transaction ("delivery-vs-payment"), remain universal challenges. The consequences of inefficiencies or interruptions in payment and settlement systems have immediate and significant impacts on the real economy. Moreover, payments and financial market infrastructures are also a matter of strategic importance for nation-states, and being excessively dependent on foreign providers has been proven to be a significant risk, as they can be "weaponized."This book provides a concise yet thorough introduction to payments and financial market infrastructures, exploring various types of risks, retail payments, payment systems, cross-border payments, central counterparties, FX operations, and central securities depositories. The final chapter investigates payment innovations, including unbacked crypto-assets (e.g. Bitcoin), stablecoins, and central bank digital currencies. The text outlines key activities and risk management frameworks, as well as examining the operational mechanics of different payment systems. The book emphasizes the importance of understanding the nature of transfer of financial claims and liabilities through the process of payment and settlement, providing financial account representations of payments wherever useful. The book concludes that the payment and financial market infrastructure space has been shaken by recent innovations, which are expected to lead to seismic shifts in payment technology, practices, and market structures.