In this paper, we first document the growing importance of foreign-domiciled investment funds in countries' portfolio liabilities over time and then show empirical evidence that cross-border fund flows are coincident with asset price movements. To measure the external liabilities of countries to foreign-domiciled funds, we complement conventional balance of payments and international investment position data with granular and real-time fund flows data. We find that the external exposure of countries to investment funds has been steadily increasing both for advanced and emerging market economies. Furthermore, we find that this increased external exposure is coincident with higher exchange rate fluctuations, lower bond yields and higher stock returns. Because sustainability-themed investment funds are growing faster than conventional investment funds, we also focus on Environmental, Social and Governance (ESG) funds and construct an index of sustainable finance that can distinguish between its domestic and cross-border components. Our index reveals that ESG funds domiciled in European countries tend to invest predominantly in domestic markets, whereas ESG investment in emerging market economies to a large extent originates from foreign-domiciled investment funds.